/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz

! This file was ported from Lean 3 source module category_theory.limits.shapes.multiequalizer
! leanprover-community/mathlib commit 97eab48559068f3d6313da387714ef25768fb730
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Shapes.Products
import Mathbin.CategoryTheory.Limits.Shapes.Equalizers
import Mathbin.CategoryTheory.Limits.ConeCategory

/-!

# Multi-(co)equalizers

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

A *multiequalizer* is an equalizer of two morphisms between two products.
Since both products and equalizers are limits, such an object is again a limit.
This file provides the diagram whose limit is indeed such an object.
In fact, it is well-known that any limit can be obtained as a multiequalizer.
The dual construction (multicoequalizers) is also provided.

## Projects

Prove that a multiequalizer can be identified with
an equalizer between products (and analogously for multicoequalizers).

Prove that the limit of any diagram is a multiequalizer (and similarly for colimits).

-/

namespace CategoryTheory.Limits

open CategoryTheory

universe w v u

#print CategoryTheory.Limits.WalkingMulticospan /-
/-- The type underlying the multiequalizer diagram. -/
@[nolint unused_arguments]
inductive WalkingMulticospan {L R : Type w} (fst snd : R → L) : Type w
  | left : L → walking_multicospan
  | right : R → walking_multicospan
#align category_theory.limits.walking_multicospan CategoryTheory.Limits.WalkingMulticospan
-/

#print CategoryTheory.Limits.WalkingMultispan /-
/-- The type underlying the multiecoqualizer diagram. -/
@[nolint unused_arguments]
inductive WalkingMultispan {L R : Type w} (fst snd : L → R) : Type w
  | left : L → walking_multispan
  | right : R → walking_multispan
#align category_theory.limits.walking_multispan CategoryTheory.Limits.WalkingMultispan
-/

namespace WalkingMulticospan

variable {L R : Type w} {fst snd : R → L}

instance [Inhabited L] : Inhabited (WalkingMulticospan fst snd) :=
  ⟨left default⟩

#print CategoryTheory.Limits.WalkingMulticospan.Hom /-
/-- Morphisms for `walking_multicospan`. -/
inductive Hom : ∀ a b : WalkingMulticospan fst snd, Type w
  | id (A) : hom A A
  | fst (b) : hom (left (fst b)) (right b)
  | snd (b) : hom (left (snd b)) (right b)
#align category_theory.limits.walking_multicospan.hom CategoryTheory.Limits.WalkingMulticospan.Hom
-/

instance {a : WalkingMulticospan fst snd} : Inhabited (Hom a a) :=
  ⟨Hom.id _⟩

#print CategoryTheory.Limits.WalkingMulticospan.Hom.comp /-
/-- Composition of morphisms for `walking_multicospan`. -/
def Hom.comp : ∀ {A B C : WalkingMulticospan fst snd} (f : Hom A B) (g : Hom B C), Hom A C
  | _, _, _, hom.id X, f => f
  | _, _, _, hom.fst b, hom.id X => Hom.fst b
  | _, _, _, hom.snd b, hom.id X => Hom.snd b
#align category_theory.limits.walking_multicospan.hom.comp CategoryTheory.Limits.WalkingMulticospan.Hom.comp
-/

instance : SmallCategory (WalkingMulticospan fst snd)
    where
  Hom := Hom
  id := Hom.id
  comp X Y Z := Hom.comp
  id_comp' := by
    rintro (_ | _) (_ | _) (_ | _ | _)
    tidy
  comp_id' := by
    rintro (_ | _) (_ | _) (_ | _ | _)
    tidy
  assoc' := by
    rintro (_ | _) (_ | _) (_ | _) (_ | _) (_ | _ | _) (_ | _ | _) (_ | _ | _)
    tidy

end WalkingMulticospan

namespace WalkingMultispan

variable {L R : Type v} {fst snd : L → R}

instance [Inhabited L] : Inhabited (WalkingMultispan fst snd) :=
  ⟨left default⟩

#print CategoryTheory.Limits.WalkingMultispan.Hom /-
/-- Morphisms for `walking_multispan`. -/
inductive Hom : ∀ a b : WalkingMultispan fst snd, Type v
  | id (A) : hom A A
  | fst (a) : hom (left a) (right (fst a))
  | snd (a) : hom (left a) (right (snd a))
#align category_theory.limits.walking_multispan.hom CategoryTheory.Limits.WalkingMultispan.Hom
-/

instance {a : WalkingMultispan fst snd} : Inhabited (Hom a a) :=
  ⟨Hom.id _⟩

#print CategoryTheory.Limits.WalkingMultispan.Hom.comp /-
/-- Composition of morphisms for `walking_multispan`. -/
def Hom.comp : ∀ {A B C : WalkingMultispan fst snd} (f : Hom A B) (g : Hom B C), Hom A C
  | _, _, _, hom.id X, f => f
  | _, _, _, hom.fst a, hom.id X => Hom.fst a
  | _, _, _, hom.snd a, hom.id X => Hom.snd a
#align category_theory.limits.walking_multispan.hom.comp CategoryTheory.Limits.WalkingMultispan.Hom.comp
-/

instance : SmallCategory (WalkingMultispan fst snd)
    where
  Hom := Hom
  id := Hom.id
  comp X Y Z := Hom.comp
  id_comp' := by
    rintro (_ | _) (_ | _) (_ | _ | _)
    tidy
  comp_id' := by
    rintro (_ | _) (_ | _) (_ | _ | _)
    tidy
  assoc' := by
    rintro (_ | _) (_ | _) (_ | _) (_ | _) (_ | _ | _) (_ | _ | _) (_ | _ | _)
    tidy

end WalkingMultispan

#print CategoryTheory.Limits.MulticospanIndex /-
/-- This is a structure encapsulating the data necessary to define a `multicospan`. -/
@[nolint has_nonempty_instance]
structure MulticospanIndex (C : Type u) [Category.{v} C] where
  (L R : Type w)
  (fstTo sndTo : R → L)
  left : L → C
  right : R → C
  fst : ∀ b, left (fst_to b) ⟶ right b
  snd : ∀ b, left (snd_to b) ⟶ right b
#align category_theory.limits.multicospan_index CategoryTheory.Limits.MulticospanIndex
-/

#print CategoryTheory.Limits.MultispanIndex /-
/-- This is a structure encapsulating the data necessary to define a `multispan`. -/
@[nolint has_nonempty_instance]
structure MultispanIndex (C : Type u) [Category.{v} C] where
  (L R : Type w)
  (fstFrom sndFrom : L → R)
  left : L → C
  right : R → C
  fst : ∀ a, left a ⟶ right (fst_from a)
  snd : ∀ a, left a ⟶ right (snd_from a)
#align category_theory.limits.multispan_index CategoryTheory.Limits.MultispanIndex
-/

namespace MulticospanIndex

variable {C : Type u} [Category.{v} C] (I : MulticospanIndex C)

#print CategoryTheory.Limits.MulticospanIndex.multicospan /-
/-- The multicospan associated to `I : multicospan_index`. -/
def multicospan : WalkingMulticospan I.fstTo I.sndTo ⥤ C
    where
  obj x :=
    match x with
    | walking_multicospan.left a => I.left a
    | walking_multicospan.right b => I.right b
  map x y f :=
    match x, y, f with
    | _, _, walking_multicospan.hom.id x => 𝟙 _
    | _, _, walking_multicospan.hom.fst b => I.fst _
    | _, _, walking_multicospan.hom.snd b => I.snd _
  map_id' := by
    rintro (_ | _)
    tidy
  map_comp' := by
    rintro (_ | _) (_ | _) (_ | _) (_ | _ | _) (_ | _ | _)
    tidy
#align category_theory.limits.multicospan_index.multicospan CategoryTheory.Limits.MulticospanIndex.multicospan
-/

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@[simp]
theorem multicospan_obj_left (a) : I.multicospan.obj (WalkingMulticospan.left a) = I.left a :=
  rfl
#align category_theory.limits.multicospan_index.multicospan_obj_left CategoryTheory.Limits.MulticospanIndex.multicospan_obj_left

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@[simp]
theorem multicospan_obj_right (b) : I.multicospan.obj (WalkingMulticospan.right b) = I.right b :=
  rfl
#align category_theory.limits.multicospan_index.multicospan_obj_right CategoryTheory.Limits.MulticospanIndex.multicospan_obj_right

/- warning: category_theory.limits.multicospan_index.multicospan_map_fst -> CategoryTheory.Limits.MulticospanIndex.multicospan_map_fst is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicospan_index.multicospan_map_fst CategoryTheory.Limits.MulticospanIndex.multicospan_map_fstₓ'. -/
@[simp]
theorem multicospan_map_fst (b) : I.multicospan.map (WalkingMulticospan.Hom.fst b) = I.fst b :=
  rfl
#align category_theory.limits.multicospan_index.multicospan_map_fst CategoryTheory.Limits.MulticospanIndex.multicospan_map_fst

/- warning: category_theory.limits.multicospan_index.multicospan_map_snd -> CategoryTheory.Limits.MulticospanIndex.multicospan_map_snd is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicospan_index.multicospan_map_snd CategoryTheory.Limits.MulticospanIndex.multicospan_map_sndₓ'. -/
@[simp]
theorem multicospan_map_snd (b) : I.multicospan.map (WalkingMulticospan.Hom.snd b) = I.snd b :=
  rfl
#align category_theory.limits.multicospan_index.multicospan_map_snd CategoryTheory.Limits.MulticospanIndex.multicospan_map_snd

variable [HasProduct I.left] [HasProduct I.right]

#print CategoryTheory.Limits.MulticospanIndex.fstPiMap /-
/-- The induced map `∏ I.left ⟶ ∏ I.right` via `I.fst`. -/
noncomputable def fstPiMap : ∏ I.left ⟶ ∏ I.right :=
  Pi.lift fun b => Pi.π I.left (I.fstTo b) ≫ I.fst b
#align category_theory.limits.multicospan_index.fst_pi_map CategoryTheory.Limits.MulticospanIndex.fstPiMap
-/

#print CategoryTheory.Limits.MulticospanIndex.sndPiMap /-
/-- The induced map `∏ I.left ⟶ ∏ I.right` via `I.snd`. -/
noncomputable def sndPiMap : ∏ I.left ⟶ ∏ I.right :=
  Pi.lift fun b => Pi.π I.left (I.sndTo b) ≫ I.snd b
#align category_theory.limits.multicospan_index.snd_pi_map CategoryTheory.Limits.MulticospanIndex.sndPiMap
-/

/- warning: category_theory.limits.multicospan_index.fst_pi_map_π -> CategoryTheory.Limits.MulticospanIndex.fstPiMap_π is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I)] (b : CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Pi.π.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3 b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.Pi.π.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2 (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.fst.{u3, u1, u2} C _inst_1 I b))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I)] (b : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Pi.π.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3 b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.Pi.π.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2 (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I b))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicospan_index.fst_pi_map_π CategoryTheory.Limits.MulticospanIndex.fstPiMap_πₓ'. -/
@[simp, reassoc.1]
theorem fstPiMap_π (b) : I.fstPiMap ≫ Pi.π I.right b = Pi.π I.left _ ≫ I.fst b := by
  simp [fst_pi_map]
#align category_theory.limits.multicospan_index.fst_pi_map_π CategoryTheory.Limits.MulticospanIndex.fstPiMap_π

/- warning: category_theory.limits.multicospan_index.snd_pi_map_π -> CategoryTheory.Limits.MulticospanIndex.sndPiMap_π is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I)] (b : CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Pi.π.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3 b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.Pi.π.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2 (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.snd.{u3, u1, u2} C _inst_1 I b))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I)] (b : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Pi.π.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3 b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.Pi.π.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2 (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I b))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicospan_index.snd_pi_map_π CategoryTheory.Limits.MulticospanIndex.sndPiMap_πₓ'. -/
@[simp, reassoc.1]
theorem sndPiMap_π (b) : I.sndPiMap ≫ Pi.π I.right b = Pi.π I.left _ ≫ I.snd b := by
  simp [snd_pi_map]
#align category_theory.limits.multicospan_index.snd_pi_map_π CategoryTheory.Limits.MulticospanIndex.sndPiMap_π

#print CategoryTheory.Limits.MulticospanIndex.parallelPairDiagram /-
/-- Taking the multiequalizer over the multicospan index is equivalent to taking the equalizer over
the two morphsims `∏ I.left ⇉ ∏ I.right`. This is the diagram of the latter.
-/
@[simps]
protected noncomputable def parallelPairDiagram :=
  parallelPair I.fstPiMap I.sndPiMap
#align category_theory.limits.multicospan_index.parallel_pair_diagram CategoryTheory.Limits.MulticospanIndex.parallelPairDiagram
-/

end MulticospanIndex

namespace MultispanIndex

variable {C : Type u} [Category.{v} C] (I : MultispanIndex C)

#print CategoryTheory.Limits.MultispanIndex.multispan /-
/-- The multispan associated to `I : multispan_index`. -/
def multispan : WalkingMultispan I.fstFrom I.sndFrom ⥤ C
    where
  obj x :=
    match x with
    | walking_multispan.left a => I.left a
    | walking_multispan.right b => I.right b
  map x y f :=
    match x, y, f with
    | _, _, walking_multispan.hom.id x => 𝟙 _
    | _, _, walking_multispan.hom.fst b => I.fst _
    | _, _, walking_multispan.hom.snd b => I.snd _
  map_id' := by
    rintro (_ | _)
    tidy
  map_comp' := by
    rintro (_ | _) (_ | _) (_ | _) (_ | _ | _) (_ | _ | _)
    tidy
#align category_theory.limits.multispan_index.multispan CategoryTheory.Limits.MultispanIndex.multispan
-/

/- warning: category_theory.limits.multispan_index.multispan_obj_left -> CategoryTheory.Limits.MultispanIndex.multispan_obj_left is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) (a : CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I), Eq.{succ u2} C (CategoryTheory.Functor.obj.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMultispan.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.WalkingMultispan.left.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I) a)) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I a)
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.multispan_obj_left CategoryTheory.Limits.MultispanIndex.multispan_obj_leftₓ'. -/
@[simp]
theorem multispan_obj_left (a) : I.multispan.obj (WalkingMultispan.left a) = I.left a :=
  rfl
#align category_theory.limits.multispan_index.multispan_obj_left CategoryTheory.Limits.MultispanIndex.multispan_obj_left

/- warning: category_theory.limits.multispan_index.multispan_obj_right -> CategoryTheory.Limits.MultispanIndex.multispan_obj_right is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) (b : CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I), Eq.{succ u2} C (CategoryTheory.Functor.obj.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMultispan.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.WalkingMultispan.right.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I) b)) (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I b)
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MultispanIndex.{u1, u2, u3} C _inst_1) (b : CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u3} C (Prefunctor.obj.{succ u1, succ u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)))) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I) b)) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I b)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.multispan_obj_right CategoryTheory.Limits.MultispanIndex.multispan_obj_rightₓ'. -/
@[simp]
theorem multispan_obj_right (b) : I.multispan.obj (WalkingMultispan.right b) = I.right b :=
  rfl
#align category_theory.limits.multispan_index.multispan_obj_right CategoryTheory.Limits.MultispanIndex.multispan_obj_right

/- warning: category_theory.limits.multispan_index.multispan_map_fst -> CategoryTheory.Limits.MultispanIndex.multispan_map_fst is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.multispan_map_fst CategoryTheory.Limits.MultispanIndex.multispan_map_fstₓ'. -/
@[simp]
theorem multispan_map_fst (a) : I.multispan.map (WalkingMultispan.Hom.fst a) = I.fst a :=
  rfl
#align category_theory.limits.multispan_index.multispan_map_fst CategoryTheory.Limits.MultispanIndex.multispan_map_fst

/- warning: category_theory.limits.multispan_index.multispan_map_snd -> CategoryTheory.Limits.MultispanIndex.multispan_map_snd is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.multispan_map_snd CategoryTheory.Limits.MultispanIndex.multispan_map_sndₓ'. -/
@[simp]
theorem multispan_map_snd (a) : I.multispan.map (WalkingMultispan.Hom.snd a) = I.snd a :=
  rfl
#align category_theory.limits.multispan_index.multispan_map_snd CategoryTheory.Limits.MultispanIndex.multispan_map_snd

variable [HasCoproduct I.left] [HasCoproduct I.right]

#print CategoryTheory.Limits.MultispanIndex.fstSigmaMap /-
/-- The induced map `∐ I.left ⟶ ∐ I.right` via `I.fst`. -/
noncomputable def fstSigmaMap : ∐ I.left ⟶ ∐ I.right :=
  Sigma.desc fun b => I.fst b ≫ Sigma.ι _ (I.fstFrom b)
#align category_theory.limits.multispan_index.fst_sigma_map CategoryTheory.Limits.MultispanIndex.fstSigmaMap
-/

#print CategoryTheory.Limits.MultispanIndex.sndSigmaMap /-
/-- The induced map `∐ I.left ⟶ ∐ I.right` via `I.snd`. -/
noncomputable def sndSigmaMap : ∐ I.left ⟶ ∐ I.right :=
  Sigma.desc fun b => I.snd b ≫ Sigma.ι _ (I.sndFrom b)
#align category_theory.limits.multispan_index.snd_sigma_map CategoryTheory.Limits.MultispanIndex.sndSigmaMap
-/

/- warning: category_theory.limits.multispan_index.ι_fst_sigma_map -> CategoryTheory.Limits.MultispanIndex.ι_fstSigmaMap is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I)] (b : CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.Sigma.ι.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2 b) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fst.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.Sigma.ι.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3 (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I b)))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.ι_fst_sigma_map CategoryTheory.Limits.MultispanIndex.ι_fstSigmaMapₓ'. -/
@[simp, reassoc.1]
theorem ι_fstSigmaMap (b) : Sigma.ι I.left b ≫ I.fstSigmaMap = I.fst b ≫ Sigma.ι I.right _ := by
  simp [fst_sigma_map]
#align category_theory.limits.multispan_index.ι_fst_sigma_map CategoryTheory.Limits.MultispanIndex.ι_fstSigmaMap

/- warning: category_theory.limits.multispan_index.ι_snd_sigma_map -> CategoryTheory.Limits.MultispanIndex.ι_sndSigmaMap is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I)] (b : CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.Sigma.ι.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2 b) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.snd.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.Sigma.ι.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3 (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I b)))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.ι_snd_sigma_map CategoryTheory.Limits.MultispanIndex.ι_sndSigmaMapₓ'. -/
@[simp, reassoc.1]
theorem ι_sndSigmaMap (b) : Sigma.ι I.left b ≫ I.sndSigmaMap = I.snd b ≫ Sigma.ι I.right _ := by
  simp [snd_sigma_map]
#align category_theory.limits.multispan_index.ι_snd_sigma_map CategoryTheory.Limits.MultispanIndex.ι_sndSigmaMap

#print CategoryTheory.Limits.MultispanIndex.parallelPairDiagram /-
/--
Taking the multicoequalizer over the multispan index is equivalent to taking the coequalizer over
the two morphsims `∐ I.left ⇉ ∐ I.right`. This is the diagram of the latter.
-/
protected noncomputable abbrev parallelPairDiagram :=
  parallelPair I.fstSigmaMap I.sndSigmaMap
#align category_theory.limits.multispan_index.parallel_pair_diagram CategoryTheory.Limits.MultispanIndex.parallelPairDiagram
-/

end MultispanIndex

variable {C : Type u} [Category.{v} C]

#print CategoryTheory.Limits.Multifork /-
/-- A multifork is a cone over a multicospan. -/
@[nolint has_nonempty_instance]
abbrev Multifork (I : MulticospanIndex C) :=
  Cone I.multicospan
#align category_theory.limits.multifork CategoryTheory.Limits.Multifork
-/

#print CategoryTheory.Limits.Multicofork /-
/-- A multicofork is a cocone over a multispan. -/
@[nolint has_nonempty_instance]
abbrev Multicofork (I : MultispanIndex C) :=
  Cocone I.multispan
#align category_theory.limits.multicofork CategoryTheory.Limits.Multicofork
-/

namespace Multifork

variable {I : MulticospanIndex C} (K : Multifork I)

#print CategoryTheory.Limits.Multifork.ι /-
/-- The maps from the cone point of a multifork to the objects on the left. -/
def ι (a : I.L) : K.pt ⟶ I.left a :=
  K.π.app (WalkingMulticospan.left _)
#align category_theory.limits.multifork.ι CategoryTheory.Limits.Multifork.ι
-/

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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.app_left_eq_ι CategoryTheory.Limits.Multifork.app_left_eq_ιₓ'. -/
@[simp]
theorem app_left_eq_ι (a) : K.π.app (WalkingMulticospan.left a) = K.ι a :=
  rfl
#align category_theory.limits.multifork.app_left_eq_ι CategoryTheory.Limits.Multifork.app_left_eq_ι

/- warning: category_theory.limits.multifork.app_right_eq_ι_comp_fst -> CategoryTheory.Limits.Multifork.app_right_eq_ι_comp_fst is a dubious translation:
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(CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.multicospan.{u1, u2, u3} C _inst_1 I) K)) (CategoryTheory.Limits.WalkingMulticospan.right.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I) b)) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Functor.obj.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMulticospan.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, 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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.app_right_eq_ι_comp_fst CategoryTheory.Limits.Multifork.app_right_eq_ι_comp_fstₓ'. -/
@[simp]
theorem app_right_eq_ι_comp_fst (b) :
    K.π.app (WalkingMulticospan.right b) = K.ι (I.fstTo b) ≫ I.fst b :=
  by
  rw [← K.w (walking_multicospan.hom.fst b)]
  rfl
#align category_theory.limits.multifork.app_right_eq_ι_comp_fst CategoryTheory.Limits.Multifork.app_right_eq_ι_comp_fst

/- warning: category_theory.limits.multifork.app_right_eq_ι_comp_snd -> CategoryTheory.Limits.Multifork.app_right_eq_ι_comp_snd is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.app_right_eq_ι_comp_snd CategoryTheory.Limits.Multifork.app_right_eq_ι_comp_sndₓ'. -/
@[reassoc.1]
theorem app_right_eq_ι_comp_snd (b) :
    K.π.app (WalkingMulticospan.right b) = K.ι (I.sndTo b) ≫ I.snd b :=
  by
  rw [← K.w (walking_multicospan.hom.snd b)]
  rfl
#align category_theory.limits.multifork.app_right_eq_ι_comp_snd CategoryTheory.Limits.Multifork.app_right_eq_ι_comp_snd

/- warning: category_theory.limits.multifork.hom_comp_ι -> CategoryTheory.Limits.Multifork.hom_comp_ι is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.hom_comp_ι CategoryTheory.Limits.Multifork.hom_comp_ιₓ'. -/
@[simp, reassoc.1]
theorem hom_comp_ι (K₁ K₂ : Multifork I) (f : K₁ ⟶ K₂) (j : I.L) : f.Hom ≫ K₂.ι j = K₁.ι j :=
  f.w (WalkingMulticospan.left j)
#align category_theory.limits.multifork.hom_comp_ι CategoryTheory.Limits.Multifork.hom_comp_ι

#print CategoryTheory.Limits.Multifork.ofι /-
/-- Construct a multifork using a collection `ι` of morphisms. -/
@[simps]
def ofι (I : MulticospanIndex C) (P : C) (ι : ∀ a, P ⟶ I.left a)
    (w : ∀ b, ι (I.fstTo b) ≫ I.fst b = ι (I.sndTo b) ≫ I.snd b) : Multifork I
    where
  pt := P
  π :=
    { app := fun x =>
        match x with
        | walking_multicospan.left a => ι _
        | walking_multicospan.right b => ι (I.fstTo b) ≫ I.fst b
      naturality' := by
        rintro (_ | _) (_ | _) (_ | _ | _)
        any_goals symm; dsimp; rw [category.id_comp]; apply category.comp_id
        · dsimp
          rw [category.id_comp]
          rfl
        · dsimp
          rw [category.id_comp]
          apply w }
#align category_theory.limits.multifork.of_ι CategoryTheory.Limits.Multifork.ofι
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.condition CategoryTheory.Limits.Multifork.conditionₓ'. -/
@[simp, reassoc.1]
theorem condition (b) : K.ι (I.fstTo b) ≫ I.fst b = K.ι (I.sndTo b) ≫ I.snd b := by
  rw [← app_right_eq_ι_comp_fst, ← app_right_eq_ι_comp_snd]
#align category_theory.limits.multifork.condition CategoryTheory.Limits.Multifork.condition

#print CategoryTheory.Limits.Multifork.IsLimit.mk /-
/-- This definition provides a convenient way to show that a multifork is a limit. -/
@[simps]
def IsLimit.mk (lift : ∀ E : Multifork I, E.pt ⟶ K.pt)
    (fac : ∀ (E : Multifork I) (i : I.L), lift E ≫ K.ι i = E.ι i)
    (uniq : ∀ (E : Multifork I) (m : E.pt ⟶ K.pt), (∀ i : I.L, m ≫ K.ι i = E.ι i) → m = lift E) :
    IsLimit K :=
  { lift
    fac := by
      rintro E (a | b)
      · apply fac
      · rw [← E.w (walking_multicospan.hom.fst b), ← K.w (walking_multicospan.hom.fst b), ←
          category.assoc]
        congr 1
        apply fac
    uniq := by
      rintro E m hm
      apply uniq
      intro i
      apply hm }
#align category_theory.limits.multifork.is_limit.mk CategoryTheory.Limits.Multifork.IsLimit.mk
-/

variable [HasProduct I.left] [HasProduct I.right]

/- warning: category_theory.limits.multifork.pi_condition -> CategoryTheory.Limits.Multifork.pi_condition is a dubious translation:
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but is expected to have type
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(CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u1} C _inst_1 I) K) (CategoryTheory.Limits.Multifork.ι.{u2, u3, u1} C _inst_1 I K)) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.pi_condition CategoryTheory.Limits.Multifork.pi_conditionₓ'. -/
/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `discrete_cases #[] -/
@[simp, reassoc.1]
theorem pi_condition : Pi.lift K.ι ≫ I.fstPiMap = Pi.lift K.ι ≫ I.sndPiMap :=
  by
  ext
  trace
    "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `discrete_cases #[]"
  simp
#align category_theory.limits.multifork.pi_condition CategoryTheory.Limits.Multifork.pi_condition

#print CategoryTheory.Limits.Multifork.toPiFork /-
/-- Given a multifork, we may obtain a fork over `∏ I.left ⇉ ∏ I.right`. -/
@[simps pt]
noncomputable def toPiFork (K : Multifork I) : Fork I.fstPiMap I.sndPiMap
    where
  pt := K.pt
  π :=
    { app := fun x =>
        match x with
        | walking_parallel_pair.zero => Pi.lift K.ι
        | walking_parallel_pair.one => Pi.lift K.ι ≫ I.fstPiMap
      naturality' := by
        rintro (_ | _) (_ | _) (_ | _ | _)
        any_goals symm; dsimp; rw [category.id_comp]; apply category.comp_id
        all_goals change 𝟙 _ ≫ _ ≫ _ = pi.lift _ ≫ _; simp }
#align category_theory.limits.multifork.to_pi_fork CategoryTheory.Limits.Multifork.toPiFork
-/

/- warning: category_theory.limits.multifork.to_pi_fork_π_app_zero -> CategoryTheory.Limits.Multifork.toPiFork_π_app_zero is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1} (K : CategoryTheory.Limits.Multifork.{u1, u2, u3} C _inst_1 I) [_inst_2 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I)], Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 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(CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K))) CategoryTheory.Limits.WalkingParallelPair.zero) (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) CategoryTheory.Limits.WalkingParallelPair.zero)) (CategoryTheory.Limits.Fork.ι.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K)) (CategoryTheory.Limits.Pi.lift.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2 (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K))) CategoryTheory.Limits.WalkingParallelPair.zero) (CategoryTheory.Limits.Multifork.ι.{u1, u2, u3} C _inst_1 I K))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] {I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1} (K : CategoryTheory.Limits.Multifork.{u2, u3, u1} C _inst_1 I) [_inst_2 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I)], Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (Prefunctor.obj.{1, succ u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.CategoryStruct.toQuiver.{0, 0} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.Category.toCategoryStruct.{0, 0} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (Prefunctor.obj.{succ u2, succ u2, u3, max u2 u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, u2, u3, max u3 u2} C _inst_1 (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1)) (CategoryTheory.Limits.Cone.pt.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 K)))) CategoryTheory.Limits.WalkingParallelPair.zero) (Prefunctor.obj.{1, succ u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.CategoryStruct.toQuiver.{0, 0} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.Category.toCategoryStruct.{0, 0} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3))) CategoryTheory.Limits.WalkingParallelPair.zero)) (CategoryTheory.Limits.Fork.ι.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Multifork.toPiFork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 K)) (CategoryTheory.Limits.Pi.lift.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (fun (b : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) => CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I b) _inst_2 (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u1} C _inst_1 I) K) (CategoryTheory.Limits.Multifork.ι.{u2, u3, u1} C _inst_1 I K))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.to_pi_fork_π_app_zero CategoryTheory.Limits.Multifork.toPiFork_π_app_zeroₓ'. -/
@[simp]
theorem toPiFork_π_app_zero : K.toPiFork.ι = Pi.lift K.ι :=
  rfl
#align category_theory.limits.multifork.to_pi_fork_π_app_zero CategoryTheory.Limits.Multifork.toPiFork_π_app_zero

/- warning: category_theory.limits.multifork.to_pi_fork_π_app_one -> CategoryTheory.Limits.Multifork.toPiFork_π_app_one is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1} (K : CategoryTheory.Limits.Multifork.{u1, u2, u3} C _inst_1 I) [_inst_2 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I)], Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 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(CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K))) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) 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CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K))) (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Cone.π.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K)) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K))) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (fun (b : CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) => CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I b) _inst_2) (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.Limits.Pi.lift.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (fun (b : CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) => CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I b) _inst_2 (CategoryTheory.Functor.obj.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 K))) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.Limits.Multifork.ι.{u1, u2, u3} C _inst_1 I K)) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] {I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1} (K : CategoryTheory.Limits.Multifork.{u2, u3, u1} C _inst_1 I) [_inst_2 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I)], Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (Prefunctor.obj.{1, succ u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.CategoryStruct.toQuiver.{0, 0} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.Category.toCategoryStruct.{0, 0} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (Prefunctor.obj.{succ u2, succ u2, u3, max u2 u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, u2, u3, max u3 u2} C _inst_1 (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1)) (CategoryTheory.Limits.Cone.pt.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 K)))) CategoryTheory.Limits.WalkingParallelPair.one) (Prefunctor.obj.{1, succ u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.CategoryStruct.toQuiver.{0, 0} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.Category.toCategoryStruct.{0, 0} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3))) CategoryTheory.Limits.WalkingParallelPair.one)) (CategoryTheory.NatTrans.app.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (Prefunctor.obj.{succ u2, succ u2, u3, max u2 u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, u2, u3, max u3 u2} C _inst_1 (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1)) (CategoryTheory.Limits.Cone.pt.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 K))) (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Cone.π.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Multifork.toPiFork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 K)) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u1} C _inst_1 I) K) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (fun (b : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) => CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I b) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.Pi.lift.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (fun (b : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) => CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I b) _inst_2 (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C 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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.to_pi_fork_π_app_one CategoryTheory.Limits.Multifork.toPiFork_π_app_oneₓ'. -/
@[simp]
theorem toPiFork_π_app_one : K.toPiFork.π.app WalkingParallelPair.one = Pi.lift K.ι ≫ I.fstPiMap :=
  rfl
#align category_theory.limits.multifork.to_pi_fork_π_app_one CategoryTheory.Limits.Multifork.toPiFork_π_app_one

variable (I)

#print CategoryTheory.Limits.Multifork.ofPiFork /-
/-- Given a fork over `∏ I.left ⇉ ∏ I.right`, we may obtain a multifork. -/
@[simps pt]
noncomputable def ofPiFork (c : Fork I.fstPiMap I.sndPiMap) : Multifork I
    where
  pt := c.pt
  π :=
    { app := fun x =>
        match x with
        | walking_multicospan.left a => c.ι ≫ Pi.π _ _
        | walking_multicospan.right b => c.ι ≫ I.fstPiMap ≫ Pi.π _ _
      naturality' := by
        rintro (_ | _) (_ | _) (_ | _ | _)
        any_goals symm; dsimp; rw [category.id_comp]; apply category.comp_id
        · change 𝟙 _ ≫ _ ≫ _ = (_ ≫ _) ≫ _
          simp
        · change 𝟙 _ ≫ _ ≫ _ = (_ ≫ _) ≫ _
          rw [c.condition_assoc]
          simp }
#align category_theory.limits.multifork.of_pi_fork CategoryTheory.Limits.Multifork.ofPiFork
-/

/- warning: category_theory.limits.multifork.of_pi_fork_π_app_left -> CategoryTheory.Limits.Multifork.ofPiFork_π_app_left is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.of_pi_fork_π_app_left CategoryTheory.Limits.Multifork.ofPiFork_π_app_leftₓ'. -/
@[simp]
theorem ofPiFork_π_app_left (c : Fork I.fstPiMap I.sndPiMap) (a) :
    (ofPiFork I c).ι a = c.ι ≫ Pi.π _ _ :=
  rfl
#align category_theory.limits.multifork.of_pi_fork_π_app_left CategoryTheory.Limits.Multifork.ofPiFork_π_app_left

/- warning: category_theory.limits.multifork.of_pi_fork_π_app_right -> CategoryTheory.Limits.Multifork.ofPiFork_π_app_right is a dubious translation:
lean 3 declaration is
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_inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) CategoryTheory.Limits.WalkingParallelPair.zero) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Functor.obj.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMulticospan.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.multicospan.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.WalkingMulticospan.right.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I) a)) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Pi.π.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3 a)))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I)] (c : CategoryTheory.Limits.Fork.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (a : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) 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C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3))) CategoryTheory.Limits.WalkingParallelPair.zero) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.Fork.ι.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) c) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (Prefunctor.obj.{1, succ u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.CategoryStruct.toQuiver.{0, 0} CategoryTheory.Limits.WalkingParallelPair (CategoryTheory.Category.toCategoryStruct.{0, 0} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3))) CategoryTheory.Limits.WalkingParallelPair.zero) (CategoryTheory.Limits.piObj.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Pi.π.{u1, u2, u3} (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3 a)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multifork.of_pi_fork_π_app_right CategoryTheory.Limits.Multifork.ofPiFork_π_app_rightₓ'. -/
@[simp]
theorem ofPiFork_π_app_right (c : Fork I.fstPiMap I.sndPiMap) (a) :
    (ofPiFork I c).π.app (WalkingMulticospan.right a) = c.ι ≫ I.fstPiMap ≫ Pi.π _ _ :=
  rfl
#align category_theory.limits.multifork.of_pi_fork_π_app_right CategoryTheory.Limits.Multifork.ofPiFork_π_app_right

end Multifork

namespace MulticospanIndex

variable (I : MulticospanIndex C) [HasProduct I.left] [HasProduct I.right]

attribute [local tidy] tactic.case_bash

#print CategoryTheory.Limits.MulticospanIndex.toPiForkFunctor /-
/-- `multifork.to_pi_fork` is functorial. -/
@[simps]
noncomputable def toPiForkFunctor : Multifork I ⥤ Fork I.fstPiMap I.sndPiMap
    where
  obj := Multifork.toPiFork
  map K₁ K₂ f :=
    { Hom := f.Hom
      w' := by
        rintro (_ | _)
        · ext
          dsimp
          simp
        · ext
          simp only [multifork.to_pi_fork_π_app_one, multifork.pi_condition, category.assoc]
          dsimp [snd_pi_map]
          simp }
#align category_theory.limits.multicospan_index.to_pi_fork_functor CategoryTheory.Limits.MulticospanIndex.toPiForkFunctor
-/

#print CategoryTheory.Limits.MulticospanIndex.ofPiForkFunctor /-
/-- `multifork.of_pi_fork` is functorial. -/
@[simps]
noncomputable def ofPiForkFunctor : Fork I.fstPiMap I.sndPiMap ⥤ Multifork I
    where
  obj := Multifork.ofPiFork I
  map K₁ K₂ f :=
    { Hom := f.Hom
      w' := by rintro (_ | _) <;> simp }
#align category_theory.limits.multicospan_index.of_pi_fork_functor CategoryTheory.Limits.MulticospanIndex.ofPiForkFunctor
-/

/- warning: category_theory.limits.multicospan_index.multifork_equiv_pi_fork -> CategoryTheory.Limits.MulticospanIndex.multiforkEquivPiFork is a dubious translation:
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but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasProduct.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I)], CategoryTheory.Equivalence.{u1, u1, max (max u3 u2) u1, max u2 u1} (CategoryTheory.Limits.Multifork.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.Fork.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Cone.category.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMulticospan.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u3} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.multicospan.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.Cone.category.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.piObj.{u3, u1, u2} (CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MulticospanIndex.fstPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MulticospanIndex.sndPiMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicospan_index.multifork_equiv_pi_fork CategoryTheory.Limits.MulticospanIndex.multiforkEquivPiForkₓ'. -/
/-- The category of multiforks is equivalent to the category of forks over `∏ I.left ⇉ ∏ I.right`.
It then follows from `category_theory.is_limit_of_preserves_cone_terminal` (or `reflects`) that it
preserves and reflects limit cones.
-/
@[simps]
noncomputable def multiforkEquivPiFork : Multifork I ≌ Fork I.fstPiMap I.sndPiMap
    where
  Functor := toPiForkFunctor I
  inverse := ofPiForkFunctor I
  unitIso :=
    NatIso.ofComponents
      (fun K =>
        Cones.ext (Iso.refl _)
          (by
            rintro (_ | _) <;> dsimp <;>
              simp [← fork.app_one_eq_ι_comp_left, -fork.app_one_eq_ι_comp_left]))
      fun K₁ K₂ f => by
      ext
      simp
  counitIso :=
    NatIso.ofComponents
      (fun K =>
        Fork.ext (Iso.refl _)
          (by
            ext ⟨j⟩
            dsimp
            simp))
      fun K₁ K₂ f => by
      ext
      simp
#align category_theory.limits.multicospan_index.multifork_equiv_pi_fork CategoryTheory.Limits.MulticospanIndex.multiforkEquivPiFork

end MulticospanIndex

namespace Multicofork

variable {I : MultispanIndex C} (K : Multicofork I)

#print CategoryTheory.Limits.Multicofork.π /-
/-- The maps to the cocone point of a multicofork from the objects on the right. -/
def π (b : I.R) : I.right b ⟶ K.pt :=
  K.ι.app (WalkingMultispan.right _)
#align category_theory.limits.multicofork.π CategoryTheory.Limits.Multicofork.π
-/

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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.π_eq_app_right CategoryTheory.Limits.Multicofork.π_eq_app_rightₓ'. -/
@[simp]
theorem π_eq_app_right (b) : K.ι.app (WalkingMultispan.right _) = K.π b :=
  rfl
#align category_theory.limits.multicofork.π_eq_app_right CategoryTheory.Limits.Multicofork.π_eq_app_right

/- warning: category_theory.limits.multicofork.fst_app_right -> CategoryTheory.Limits.Multicofork.fst_app_right is a dubious translation:
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_inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I) K)) (CategoryTheory.Limits.WalkingMultispan.left.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I) a)) (CategoryTheory.Limits.MultispanIndex.fst.{u3, u1, u2} C _inst_1 I a) (CategoryTheory.Limits.Multicofork.π.{u1, u2, u3} C _inst_1 I K (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I a)))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.fst_app_right CategoryTheory.Limits.Multicofork.fst_app_rightₓ'. -/
@[simp]
theorem fst_app_right (a) : K.ι.app (WalkingMultispan.left a) = I.fst a ≫ K.π _ :=
  by
  rw [← K.w (walking_multispan.hom.fst a)]
  rfl
#align category_theory.limits.multicofork.fst_app_right CategoryTheory.Limits.Multicofork.fst_app_right

/- warning: category_theory.limits.multicofork.snd_app_right -> CategoryTheory.Limits.Multicofork.snd_app_right is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.snd_app_right CategoryTheory.Limits.Multicofork.snd_app_rightₓ'. -/
@[reassoc.1]
theorem snd_app_right (a) : K.ι.app (WalkingMultispan.left a) = I.snd a ≫ K.π _ :=
  by
  rw [← K.w (walking_multispan.hom.snd a)]
  rfl
#align category_theory.limits.multicofork.snd_app_right CategoryTheory.Limits.Multicofork.snd_app_right

#print CategoryTheory.Limits.Multicofork.ofπ /-
/-- Construct a multicofork using a collection `π` of morphisms. -/
@[simps]
def ofπ (I : MultispanIndex C) (P : C) (π : ∀ b, I.right b ⟶ P)
    (w : ∀ a, I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)) : Multicofork I
    where
  pt := P
  ι :=
    { app := fun x =>
        match x with
        | walking_multispan.left a => I.fst a ≫ π _
        | walking_multispan.right b => π _
      naturality' := by
        rintro (_ | _) (_ | _) (_ | _ | _)
        any_goals dsimp; rw [category.comp_id]; apply category.id_comp
        · dsimp
          rw [category.comp_id]
          rfl
        · dsimp
          rw [category.comp_id]
          apply (w _).symm }
#align category_theory.limits.multicofork.of_π CategoryTheory.Limits.Multicofork.ofπ
-/

/- warning: category_theory.limits.multicofork.condition -> CategoryTheory.Limits.Multicofork.condition is a dubious translation:
lean 3 declaration is
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but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] {I : CategoryTheory.Limits.MultispanIndex.{u1, u2, u3} C _inst_1} (K : CategoryTheory.Limits.Multicofork.{u2, u3, u1} C _inst_1 I) (a : CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.Cocone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I) K)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I a)) (CategoryTheory.Limits.Cocone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I) K) (CategoryTheory.Limits.MultispanIndex.fst.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.Multicofork.π.{u2, u3, u1} C _inst_1 I K (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I a))) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I a)) (CategoryTheory.Limits.Cocone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I) K) (CategoryTheory.Limits.MultispanIndex.snd.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.Multicofork.π.{u2, u3, u1} C _inst_1 I K (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I a)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.condition CategoryTheory.Limits.Multicofork.conditionₓ'. -/
@[simp, reassoc.1]
theorem condition (a) : I.fst a ≫ K.π (I.fstFrom a) = I.snd a ≫ K.π (I.sndFrom a) := by
  rw [← K.snd_app_right, ← K.fst_app_right]
#align category_theory.limits.multicofork.condition CategoryTheory.Limits.Multicofork.condition

#print CategoryTheory.Limits.Multicofork.IsColimit.mk /-
/-- This definition provides a convenient way to show that a multicofork is a colimit. -/
@[simps]
def IsColimit.mk (desc : ∀ E : Multicofork I, K.pt ⟶ E.pt)
    (fac : ∀ (E : Multicofork I) (i : I.R), K.π i ≫ desc E = E.π i)
    (uniq : ∀ (E : Multicofork I) (m : K.pt ⟶ E.pt), (∀ i : I.R, K.π i ≫ m = E.π i) → m = desc E) :
    IsColimit K :=
  { desc
    fac := by
      rintro S (a | b)
      · rw [← K.w (walking_multispan.hom.fst a), ← S.w (walking_multispan.hom.fst a),
          category.assoc]
        congr 1
        apply fac
      · apply fac
    uniq := by
      intro S m hm
      apply uniq
      intro i
      apply hm }
#align category_theory.limits.multicofork.is_colimit.mk CategoryTheory.Limits.Multicofork.IsColimit.mk
-/

variable [HasCoproduct I.left] [HasCoproduct I.right]

/- warning: category_theory.limits.multicofork.sigma_condition -> CategoryTheory.Limits.Multicofork.sigma_condition is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1} (K : CategoryTheory.Limits.Multicofork.{u1, u2, u3} C _inst_1 I) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I)], Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) 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but is expected to have type
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_inst_1 I) _inst_3 (CategoryTheory.Limits.Cocone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I) K) (CategoryTheory.Limits.Multicofork.π.{u2, u3, u1} C _inst_1 I K)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.sigma_condition CategoryTheory.Limits.Multicofork.sigma_conditionₓ'. -/
/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `discrete_cases #[] -/
@[simp, reassoc.1]
theorem sigma_condition : I.fstSigmaMap ≫ Sigma.desc K.π = I.sndSigmaMap ≫ Sigma.desc K.π :=
  by
  ext
  trace
    "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `discrete_cases #[]"
  simp
#align category_theory.limits.multicofork.sigma_condition CategoryTheory.Limits.Multicofork.sigma_condition

#print CategoryTheory.Limits.Multicofork.toSigmaCofork /-
/-- Given a multicofork, we may obtain a cofork over `∐ I.left ⇉ ∐ I.right`. -/
@[simps pt]
noncomputable def toSigmaCofork (K : Multicofork I) : Cofork I.fstSigmaMap I.sndSigmaMap
    where
  pt := K.pt
  ι :=
    { app := fun x =>
        match x with
        | walking_parallel_pair.zero => I.fstSigmaMap ≫ Sigma.desc K.π
        | walking_parallel_pair.one => Sigma.desc K.π
      naturality' := by
        rintro (_ | _) (_ | _) (_ | _ | _)
        any_goals dsimp; rw [category.comp_id]; apply category.id_comp
        all_goals change _ ≫ sigma.desc _ = (_ ≫ _) ≫ 𝟙 _; simp }
#align category_theory.limits.multicofork.to_sigma_cofork CategoryTheory.Limits.Multicofork.toSigmaCofork
-/

/- warning: category_theory.limits.multicofork.to_sigma_cofork_π -> CategoryTheory.Limits.Multicofork.toSigmaCofork_π is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.to_sigma_cofork_π CategoryTheory.Limits.Multicofork.toSigmaCofork_πₓ'. -/
@[simp]
theorem toSigmaCofork_π : K.toSigmaCofork.π = Sigma.desc K.π :=
  rfl
#align category_theory.limits.multicofork.to_sigma_cofork_π CategoryTheory.Limits.Multicofork.toSigmaCofork_π

variable (I)

#print CategoryTheory.Limits.Multicofork.ofSigmaCofork /-
/-- Given a cofork over `∐ I.left ⇉ ∐ I.right`, we may obtain a multicofork. -/
@[simps pt]
noncomputable def ofSigmaCofork (c : Cofork I.fstSigmaMap I.sndSigmaMap) : Multicofork I
    where
  pt := c.pt
  ι :=
    { app := fun x =>
        match x with
        | walking_multispan.left a => (Sigma.ι I.left a : _) ≫ I.fstSigmaMap ≫ c.π
        | walking_multispan.right b => (Sigma.ι I.right b : _) ≫ c.π
      naturality' := by
        rintro (_ | _) (_ | _) (_ | _ | _)
        any_goals dsimp; rw [category.comp_id]; apply category.id_comp
        · change _ ≫ _ ≫ _ = (_ ≫ _) ≫ _
          dsimp
          simp only [cofork.condition, category.comp_id]
          rw [← I.ι_fst_sigma_map_assoc, c.condition]
        · change _ ≫ _ ≫ _ = (_ ≫ _) ≫ 𝟙 _
          rw [c.condition]
          simp }
#align category_theory.limits.multicofork.of_sigma_cofork CategoryTheory.Limits.Multicofork.ofSigmaCofork
-/

/- warning: category_theory.limits.multicofork.of_sigma_cofork_ι_app_left -> CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_left is a dubious translation:
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_inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.Multicofork.ofSigmaCofork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 c))) (CategoryTheory.Limits.WalkingMultispan.left.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I) a)) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.Cofork.π.{u1, u2} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) c)))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MultispanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I)] (c : CategoryTheory.Limits.Cofork.{u2, u3} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) (a : CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.of_sigma_cofork_ι_app_left CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_leftₓ'. -/
@[simp]
theorem ofSigmaCofork_ι_app_left (c : Cofork I.fstSigmaMap I.sndSigmaMap) (a) :
    (ofSigmaCofork I c).ι.app (WalkingMultispan.left a) =
      (Sigma.ι I.left a : _) ≫ I.fstSigmaMap ≫ c.π :=
  rfl
#align category_theory.limits.multicofork.of_sigma_cofork_ι_app_left CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_left

/- warning: category_theory.limits.multicofork.of_sigma_cofork_ι_app_right -> CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_right is a dubious translation:
lean 3 declaration is
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_inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.Multicofork.ofSigmaCofork.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 c))) (CategoryTheory.Limits.WalkingMultispan.right.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I) b)) (CategoryTheory.Limits.Sigma.ι.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3 b) (CategoryTheory.Limits.Cofork.π.{u1, u2} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) c))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MultispanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I)] (c : CategoryTheory.Limits.Cofork.{u2, u3} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) 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(CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I) (CategoryTheory.Limits.Multicofork.ofSigmaCofork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 c))) (CategoryTheory.Limits.Cocone.ι.{u1, u2, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u2, u3} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u2, u3, u1} C _inst_1 I) (CategoryTheory.Limits.Multicofork.ofSigmaCofork.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 c)) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) 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u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (Prefunctor.obj.{succ u2, succ u2, u3, max u2 u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u2, max u3 u2} (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, u2, u3, max u3 u2} C _inst_1 (CategoryTheory.Functor.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.category.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1) (CategoryTheory.Functor.const.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{0, u2, 0, u3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u2, u3} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3)) c))) CategoryTheory.Limits.WalkingParallelPair.one) (CategoryTheory.Limits.Sigma.ι.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3 b) (CategoryTheory.Limits.Cofork.π.{u2, u3} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3) c))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicofork.of_sigma_cofork_ι_app_right CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_rightₓ'. -/
@[simp]
theorem ofSigmaCofork_ι_app_right (c : Cofork I.fstSigmaMap I.sndSigmaMap) (b) :
    (ofSigmaCofork I c).ι.app (WalkingMultispan.right b) = (Sigma.ι I.right b : _) ≫ c.π :=
  rfl
#align category_theory.limits.multicofork.of_sigma_cofork_ι_app_right CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_right

end Multicofork

namespace MultispanIndex

variable (I : MultispanIndex C) [HasCoproduct I.left] [HasCoproduct I.right]

attribute [local tidy] tactic.case_bash

#print CategoryTheory.Limits.MultispanIndex.toSigmaCoforkFunctor /-
/-- `multicofork.to_sigma_cofork` is functorial. -/
@[simps]
noncomputable def toSigmaCoforkFunctor : Multicofork I ⥤ Cofork I.fstSigmaMap I.sndSigmaMap
    where
  obj := Multicofork.toSigmaCofork
  map K₁ K₂ f := { Hom := f.Hom }
#align category_theory.limits.multispan_index.to_sigma_cofork_functor CategoryTheory.Limits.MultispanIndex.toSigmaCoforkFunctor
-/

#print CategoryTheory.Limits.MultispanIndex.ofSigmaCoforkFunctor /-
/-- `multicofork.of_sigma_cofork` is functorial. -/
@[simps]
noncomputable def ofSigmaCoforkFunctor : Cofork I.fstSigmaMap I.sndSigmaMap ⥤ Multicofork I
    where
  obj := Multicofork.ofSigmaCofork I
  map K₁ K₂ f :=
    { Hom := f.Hom
      w' := by rintro (_ | _) <;> simp }
#align category_theory.limits.multispan_index.of_sigma_cofork_functor CategoryTheory.Limits.MultispanIndex.ofSigmaCoforkFunctor
-/

/- warning: category_theory.limits.multispan_index.multicofork_equiv_sigma_cofork -> CategoryTheory.Limits.MultispanIndex.multicoforkEquivSigmaCofork is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I)], CategoryTheory.Equivalence.{u1, u1, max u3 u2 u1, max u2 u1} (CategoryTheory.Limits.Multicofork.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.Cocone.category.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMultispan.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.Cofork.{u1, u2} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Cocone.category.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I)], CategoryTheory.Equivalence.{u1, u1, max (max u3 u2) u1, max u2 u1} (CategoryTheory.Limits.Multicofork.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.Cofork.{u1, u2} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)) (CategoryTheory.Limits.Cocone.category.{u3, u1, u3, u2} (CategoryTheory.Limits.WalkingMultispan.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u3} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u3, u1, u2} C _inst_1 I) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u3, u1, u2} C _inst_1 I)) C _inst_1 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.Cocone.category.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I) _inst_2) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_3) (CategoryTheory.Limits.MultispanIndex.fstSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3) (CategoryTheory.Limits.MultispanIndex.sndSigmaMap.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3)))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multispan_index.multicofork_equiv_sigma_cofork CategoryTheory.Limits.MultispanIndex.multicoforkEquivSigmaCoforkₓ'. -/
/--
The category of multicoforks is equivalent to the category of coforks over `∐ I.left ⇉ ∐ I.right`.
It then follows from `category_theory.is_colimit_of_preserves_cocone_initial` (or `reflects`) that
it preserves and reflects colimit cocones.
-/
@[simps]
noncomputable def multicoforkEquivSigmaCofork : Multicofork I ≌ Cofork I.fstSigmaMap I.sndSigmaMap
    where
  Functor := toSigmaCoforkFunctor I
  inverse := ofSigmaCoforkFunctor I
  unitIso :=
    NatIso.ofComponents (fun K => Cocones.ext (Iso.refl _) (by rintro (_ | _) <;> dsimp <;> simp))
      fun K₁ K₂ f => by
      ext
      simp
  counitIso :=
    NatIso.ofComponents
      (fun K =>
        Cofork.ext (Iso.refl _)
          (by
            ext ⟨j⟩
            dsimp
            simp only [category.comp_id, colimit.ι_desc, cofan.mk_ι_app]
            rfl))
      fun K₁ K₂ f => by
      ext
      dsimp
      simp
#align category_theory.limits.multispan_index.multicofork_equiv_sigma_cofork CategoryTheory.Limits.MultispanIndex.multicoforkEquivSigmaCofork

end MultispanIndex

#print CategoryTheory.Limits.HasMultiequalizer /-
/-- For `I : multicospan_index C`, we say that it has a multiequalizer if the associated
  multicospan has a limit. -/
abbrev HasMultiequalizer (I : MulticospanIndex C) :=
  HasLimit I.multicospan
#align category_theory.limits.has_multiequalizer CategoryTheory.Limits.HasMultiequalizer
-/

noncomputable section

#print CategoryTheory.Limits.multiequalizer /-
/-- The multiequalizer of `I : multicospan_index C`. -/
abbrev multiequalizer (I : MulticospanIndex C) [HasMultiequalizer I] : C :=
  limit I.multicospan
#align category_theory.limits.multiequalizer CategoryTheory.Limits.multiequalizer
-/

#print CategoryTheory.Limits.HasMulticoequalizer /-
/-- For `I : multispan_index C`, we say that it has a multicoequalizer if
  the associated multicospan has a limit. -/
abbrev HasMulticoequalizer (I : MultispanIndex C) :=
  HasColimit I.multispan
#align category_theory.limits.has_multicoequalizer CategoryTheory.Limits.HasMulticoequalizer
-/

#print CategoryTheory.Limits.multicoequalizer /-
/-- The multiecoqualizer of `I : multispan_index C`. -/
abbrev multicoequalizer (I : MultispanIndex C) [HasMulticoequalizer I] : C :=
  colimit I.multispan
#align category_theory.limits.multicoequalizer CategoryTheory.Limits.multicoequalizer
-/

namespace Multiequalizer

variable (I : MulticospanIndex C) [HasMultiequalizer I]

#print CategoryTheory.Limits.Multiequalizer.ι /-
/-- The canonical map from the multiequalizer to the objects on the left. -/
abbrev ι (a : I.L) : multiequalizer I ⟶ I.left a :=
  limit.π _ (WalkingMulticospan.left a)
#align category_theory.limits.multiequalizer.ι CategoryTheory.Limits.Multiequalizer.ι
-/

#print CategoryTheory.Limits.Multiequalizer.multifork /-
/-- The multifork associated to the multiequalizer. -/
abbrev multifork : Multifork I :=
  limit.cone _
#align category_theory.limits.multiequalizer.multifork CategoryTheory.Limits.Multiequalizer.multifork
-/

/- warning: category_theory.limits.multiequalizer.multifork_ι -> CategoryTheory.Limits.Multiequalizer.multifork_ι is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multiequalizer.multifork_ι CategoryTheory.Limits.Multiequalizer.multifork_ιₓ'. -/
@[simp]
theorem multifork_ι (a) : (Multiequalizer.multifork I).ι a = Multiequalizer.ι I a :=
  rfl
#align category_theory.limits.multiequalizer.multifork_ι CategoryTheory.Limits.Multiequalizer.multifork_ι

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multiequalizer.multifork_π_app_left CategoryTheory.Limits.Multiequalizer.multifork_π_app_leftₓ'. -/
@[simp]
theorem multifork_π_app_left (a) :
    (Multiequalizer.multifork I).π.app (WalkingMulticospan.left a) = Multiequalizer.ι I a :=
  rfl
#align category_theory.limits.multiequalizer.multifork_π_app_left CategoryTheory.Limits.Multiequalizer.multifork_π_app_left

/- warning: category_theory.limits.multiequalizer.condition -> CategoryTheory.Limits.Multiequalizer.condition is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMultiequalizer.{u1, u2, u3} C _inst_1 I] (b : CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.Multiequalizer.ι.{u1, u2, u3} C _inst_1 I _inst_2 (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.fst.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.Multiequalizer.ι.{u1, u2, u3} C _inst_1 I _inst_2 (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.snd.{u3, u1, u2} C _inst_1 I b))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMultiequalizer.{u2, u3, u1} C _inst_1 I] (b : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.Multiequalizer.ι.{u2, u3, u1} C _inst_1 I _inst_2 (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.Multiequalizer.ι.{u2, u3, u1} C _inst_1 I _inst_2 (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I b))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multiequalizer.condition CategoryTheory.Limits.Multiequalizer.conditionₓ'. -/
@[reassoc.1]
theorem condition (b) :
    Multiequalizer.ι I (I.fstTo b) ≫ I.fst b = Multiequalizer.ι I (I.sndTo b) ≫ I.snd b :=
  Multifork.condition _ _
#align category_theory.limits.multiequalizer.condition CategoryTheory.Limits.Multiequalizer.condition

#print CategoryTheory.Limits.Multiequalizer.lift /-
/-- Construct a morphism to the multiequalizer from its universal property. -/
abbrev lift (W : C) (k : ∀ a, W ⟶ I.left a)
    (h : ∀ b, k (I.fstTo b) ≫ I.fst b = k (I.sndTo b) ≫ I.snd b) : W ⟶ multiequalizer I :=
  limit.lift _ (Multifork.ofι I _ k h)
#align category_theory.limits.multiequalizer.lift CategoryTheory.Limits.Multiequalizer.lift
-/

/- warning: category_theory.limits.multiequalizer.lift_ι -> CategoryTheory.Limits.Multiequalizer.lift_ι is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMultiequalizer.{u1, u2, u3} C _inst_1 I] (W : C) (k : forall (a : CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I), Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I a)) (h : forall (b : CategoryTheory.Limits.MulticospanIndex.R.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) W (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (k (CategoryTheory.Limits.MulticospanIndex.fstTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.fst.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) W (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u3, u1, u2} C _inst_1 I b) (k (CategoryTheory.Limits.MulticospanIndex.sndTo.{u3, u1, u2} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.snd.{u3, u1, u2} C _inst_1 I b))) (a : CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I a)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) W (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I a) (CategoryTheory.Limits.Multiequalizer.lift.{u1, u2, u3} C _inst_1 I _inst_2 W k h) (CategoryTheory.Limits.Multiequalizer.ι.{u1, u2, u3} C _inst_1 I _inst_2 a)) (k a)
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMultiequalizer.{u2, u3, u1} C _inst_1 I] (W : C) (k : forall (a : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I a)) (h : forall (b : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) W (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (k (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) W (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I b) (k (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I b)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I b))) (a : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I a)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) W (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I a) (CategoryTheory.Limits.Multiequalizer.lift.{u2, u3, u1} C _inst_1 I _inst_2 W k h) (CategoryTheory.Limits.Multiequalizer.ι.{u2, u3, u1} C _inst_1 I _inst_2 a)) (k a)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multiequalizer.lift_ι CategoryTheory.Limits.Multiequalizer.lift_ιₓ'. -/
@[simp, reassoc.1]
theorem lift_ι (W : C) (k : ∀ a, W ⟶ I.left a)
    (h : ∀ b, k (I.fstTo b) ≫ I.fst b = k (I.sndTo b) ≫ I.snd b) (a) :
    Multiequalizer.lift I _ k h ≫ Multiequalizer.ι I a = k _ :=
  limit.lift_π _ _
#align category_theory.limits.multiequalizer.lift_ι CategoryTheory.Limits.Multiequalizer.lift_ι

/- warning: category_theory.limits.multiequalizer.hom_ext -> CategoryTheory.Limits.Multiequalizer.hom_ext is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MulticospanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMultiequalizer.{u1, u2, u3} C _inst_1 I] {W : C} (i : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2)) (j : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2)), (forall (a : CategoryTheory.Limits.MulticospanIndex.L.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I a)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) W (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I a) i (CategoryTheory.Limits.Multiequalizer.ι.{u1, u2, u3} C _inst_1 I _inst_2 a)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) W (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u3, u1, u2} C _inst_1 I a) j (CategoryTheory.Limits.Multiequalizer.ι.{u1, u2, u3} C _inst_1 I _inst_2 a))) -> (Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) W (CategoryTheory.Limits.multiequalizer.{u1, u2, u3} C _inst_1 I _inst_2)) i j)
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multiequalizer.hom_ext CategoryTheory.Limits.Multiequalizer.hom_extₓ'. -/
@[ext]
theorem hom_ext {W : C} (i j : W ⟶ multiequalizer I)
    (h : ∀ a, i ≫ Multiequalizer.ι I a = j ≫ Multiequalizer.ι I a) : i = j :=
  limit.hom_ext
    (by
      rintro (a | b)
      · apply h
      simp_rw [← limit.w I.multicospan (walking_multicospan.hom.fst b), ← category.assoc, h])
#align category_theory.limits.multiequalizer.hom_ext CategoryTheory.Limits.Multiequalizer.hom_ext

variable [HasProduct I.left] [HasProduct I.right]

instance : HasEqualizer I.fstPiMap I.sndPiMap :=
  ⟨⟨⟨_, IsLimit.ofPreservesConeTerminal I.multiforkEquivPiFork.Functor (limit.isLimit _)⟩⟩⟩

#print CategoryTheory.Limits.Multiequalizer.isoEqualizer /-
/-- The multiequalizer is isomorphic to the equalizer of `∏ I.left ⇉ ∏ I.right`. -/
def isoEqualizer : multiequalizer I ≅ equalizer I.fstPiMap I.sndPiMap :=
  limit.isoLimitCone
    ⟨_, IsLimit.ofPreservesConeTerminal I.multiforkEquivPiFork.inverse (limit.isLimit _)⟩
#align category_theory.limits.multiequalizer.iso_equalizer CategoryTheory.Limits.Multiequalizer.isoEqualizer
-/

#print CategoryTheory.Limits.Multiequalizer.ιPi /-
/-- The canonical injection `multiequalizer I ⟶ ∏ I.left`. -/
def ιPi : multiequalizer I ⟶ ∏ I.left :=
  (isoEqualizer I).Hom ≫ equalizer.ι I.fstPiMap I.sndPiMap
#align category_theory.limits.multiequalizer.ι_pi CategoryTheory.Limits.Multiequalizer.ιPi
-/

/- warning: category_theory.limits.multiequalizer.ι_pi_π -> CategoryTheory.Limits.Multiequalizer.ιPi_π is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multiequalizer.ι_pi_π CategoryTheory.Limits.Multiequalizer.ιPi_πₓ'. -/
@[simp, reassoc.1]
theorem ιPi_π (a) : ιPi I ≫ Pi.π I.left a = ι I a :=
  by
  rw [ι_pi, category.assoc, ← iso.eq_inv_comp, iso_equalizer]
  simpa
#align category_theory.limits.multiequalizer.ι_pi_π CategoryTheory.Limits.Multiequalizer.ιPi_π

instance : Mono (ιPi I) :=
  @mono_comp _ _ _ _ equalizer.ι_mono

end Multiequalizer

namespace Multicoequalizer

variable (I : MultispanIndex C) [HasMulticoequalizer I]

#print CategoryTheory.Limits.Multicoequalizer.π /-
/-- The canonical map from the multiequalizer to the objects on the left. -/
abbrev π (b : I.R) : I.right b ⟶ multicoequalizer I :=
  colimit.ι I.multispan (WalkingMultispan.right _)
#align category_theory.limits.multicoequalizer.π CategoryTheory.Limits.Multicoequalizer.π
-/

#print CategoryTheory.Limits.Multicoequalizer.multicofork /-
/-- The multicofork associated to the multicoequalizer. -/
abbrev multicofork : Multicofork I :=
  colimit.cocone _
#align category_theory.limits.multicoequalizer.multicofork CategoryTheory.Limits.Multicoequalizer.multicofork
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicoequalizer.multicofork_π CategoryTheory.Limits.Multicoequalizer.multicofork_πₓ'. -/
@[simp]
theorem multicofork_π (b) : (Multicoequalizer.multicofork I).π b = Multicoequalizer.π I b :=
  rfl
#align category_theory.limits.multicoequalizer.multicofork_π CategoryTheory.Limits.Multicoequalizer.multicofork_π

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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicoequalizer.multicofork_ι_app_right CategoryTheory.Limits.Multicoequalizer.multicofork_ι_app_rightₓ'. -/
@[simp]
theorem multicofork_ι_app_right (b) :
    (Multicoequalizer.multicofork I).ι.app (WalkingMultispan.right b) = Multicoequalizer.π I b :=
  rfl
#align category_theory.limits.multicoequalizer.multicofork_ι_app_right CategoryTheory.Limits.Multicoequalizer.multicofork_ι_app_right

/- warning: category_theory.limits.multicoequalizer.condition -> CategoryTheory.Limits.Multicoequalizer.condition is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicoequalizer.condition CategoryTheory.Limits.Multicoequalizer.conditionₓ'. -/
@[reassoc.1]
theorem condition (a) :
    I.fst a ≫ Multicoequalizer.π I (I.fstFrom a) = I.snd a ≫ Multicoequalizer.π I (I.sndFrom a) :=
  Multicofork.condition _ _
#align category_theory.limits.multicoequalizer.condition CategoryTheory.Limits.Multicoequalizer.condition

#print CategoryTheory.Limits.Multicoequalizer.desc /-
/-- Construct a morphism from the multicoequalizer from its universal property. -/
abbrev desc (W : C) (k : ∀ b, I.right b ⟶ W)
    (h : ∀ a, I.fst a ≫ k (I.fstFrom a) = I.snd a ≫ k (I.sndFrom a)) : multicoequalizer I ⟶ W :=
  colimit.desc _ (Multicofork.ofπ I _ k h)
#align category_theory.limits.multicoequalizer.desc CategoryTheory.Limits.Multicoequalizer.desc
-/

/- warning: category_theory.limits.multicoequalizer.π_desc -> CategoryTheory.Limits.Multicoequalizer.π_desc is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicoequalizer.π_desc CategoryTheory.Limits.Multicoequalizer.π_descₓ'. -/
@[simp, reassoc.1]
theorem π_desc (W : C) (k : ∀ b, I.right b ⟶ W)
    (h : ∀ a, I.fst a ≫ k (I.fstFrom a) = I.snd a ≫ k (I.sndFrom a)) (b) :
    Multicoequalizer.π I b ≫ Multicoequalizer.desc I _ k h = k _ :=
  colimit.ι_desc _ _
#align category_theory.limits.multicoequalizer.π_desc CategoryTheory.Limits.Multicoequalizer.π_desc

/- warning: category_theory.limits.multicoequalizer.hom_ext -> CategoryTheory.Limits.Multicoequalizer.hom_ext is a dubious translation:
lean 3 declaration is
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but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MultispanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMulticoequalizer.{u2, u3, u1} C _inst_1 I] {W : C} (i : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2) W) (j : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2) W), (forall (b : CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I b) W) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2) W (CategoryTheory.Limits.Multicoequalizer.π.{u2, u3, u1} C _inst_1 I _inst_2 b) i) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2) W (CategoryTheory.Limits.Multicoequalizer.π.{u2, u3, u1} C _inst_1 I _inst_2 b) j)) -> (Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2) W) i j)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicoequalizer.hom_ext CategoryTheory.Limits.Multicoequalizer.hom_extₓ'. -/
@[ext]
theorem hom_ext {W : C} (i j : multicoequalizer I ⟶ W)
    (h : ∀ b, Multicoequalizer.π I b ≫ i = Multicoequalizer.π I b ≫ j) : i = j :=
  colimit.hom_ext
    (by
      rintro (a | b)
      · simp_rw [← colimit.w I.multispan (walking_multispan.hom.fst a), category.assoc, h]
      · apply h)
#align category_theory.limits.multicoequalizer.hom_ext CategoryTheory.Limits.Multicoequalizer.hom_ext

variable [HasCoproduct I.left] [HasCoproduct I.right]

instance : HasCoequalizer I.fstSigmaMap I.sndSigmaMap :=
  ⟨⟨⟨_,
        IsColimit.ofPreservesCoconeInitial I.multicoforkEquivSigmaCofork.Functor
          (colimit.isColimit _)⟩⟩⟩

#print CategoryTheory.Limits.Multicoequalizer.isoCoequalizer /-
/-- The multicoequalizer is isomorphic to the coequalizer of `∐ I.left ⇉ ∐ I.right`. -/
def isoCoequalizer : multicoequalizer I ≅ coequalizer I.fstSigmaMap I.sndSigmaMap :=
  colimit.isoColimitCocone
    ⟨_,
      IsColimit.ofPreservesCoconeInitial I.multicoforkEquivSigmaCofork.inverse
        (colimit.isColimit _)⟩
#align category_theory.limits.multicoequalizer.iso_coequalizer CategoryTheory.Limits.Multicoequalizer.isoCoequalizer
-/

#print CategoryTheory.Limits.Multicoequalizer.sigmaπ /-
/-- The canonical projection `∐ I.right ⟶ multicoequalizer I`. -/
def sigmaπ : ∐ I.right ⟶ multicoequalizer I :=
  coequalizer.π I.fstSigmaMap I.sndSigmaMap ≫ (isoCoequalizer I).inv
#align category_theory.limits.multicoequalizer.sigma_π CategoryTheory.Limits.Multicoequalizer.sigmaπ
-/

/- warning: category_theory.limits.multicoequalizer.ι_sigma_π -> CategoryTheory.Limits.Multicoequalizer.ι_sigmaπ is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (I : CategoryTheory.Limits.MultispanIndex.{u3, u1, u2} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMulticoequalizer.{u1, u2, u3} C _inst_1 I] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.L.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u3, u1, u2} C _inst_1 I)] [_inst_4 : CategoryTheory.Limits.HasCoproduct.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I)] (b : CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.multicoequalizer.{u1, u2, u3} C _inst_1 I _inst_2)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I b) (CategoryTheory.Limits.sigmaObj.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_4) (CategoryTheory.Limits.multicoequalizer.{u1, u2, u3} C _inst_1 I _inst_2) (CategoryTheory.Limits.Sigma.ι.{u3, u1, u2} (CategoryTheory.Limits.MultispanIndex.R.{u3, u1, u2} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u3, u1, u2} C _inst_1 I) _inst_4 b) (CategoryTheory.Limits.Multicoequalizer.sigmaπ.{u1, u2, u3} C _inst_1 I _inst_2 _inst_3 _inst_4)) (CategoryTheory.Limits.Multicoequalizer.π.{u1, u2, u3} C _inst_1 I _inst_2 b)
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] (I : CategoryTheory.Limits.MultispanIndex.{u1, u2, u3} C _inst_1) [_inst_2 : CategoryTheory.Limits.HasMulticoequalizer.{u2, u3, u1} C _inst_1 I] [_inst_3 : CategoryTheory.Limits.HasCoproduct.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.L.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.left.{u1, u2, u3} C _inst_1 I)] [_inst_4 : CategoryTheory.Limits.HasCoproduct.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I)] (b : CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2)) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I b) (CategoryTheory.Limits.sigmaObj.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_4) (CategoryTheory.Limits.multicoequalizer.{u2, u3, u1} C _inst_1 I _inst_2) (CategoryTheory.Limits.Sigma.ι.{u1, u2, u3} (CategoryTheory.Limits.MultispanIndex.R.{u1, u2, u3} C _inst_1 I) C _inst_1 (CategoryTheory.Limits.MultispanIndex.right.{u1, u2, u3} C _inst_1 I) _inst_4 b) (CategoryTheory.Limits.Multicoequalizer.sigmaπ.{u2, u3, u1} C _inst_1 I _inst_2 _inst_3 _inst_4)) (CategoryTheory.Limits.Multicoequalizer.π.{u2, u3, u1} C _inst_1 I _inst_2 b)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.multicoequalizer.ι_sigma_π CategoryTheory.Limits.Multicoequalizer.ι_sigmaπₓ'. -/
@[simp, reassoc.1]
theorem ι_sigmaπ (b) : Sigma.ι I.right b ≫ sigmaπ I = π I b :=
  by
  rw [sigma_π, ← category.assoc, iso.comp_inv_eq, iso_coequalizer]
  simpa
#align category_theory.limits.multicoequalizer.ι_sigma_π CategoryTheory.Limits.Multicoequalizer.ι_sigmaπ

instance : Epi (sigmaπ I) :=
  @epi_comp _ _ coequalizer.π_epi _ _

end Multicoequalizer

end CategoryTheory.Limits