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

! This file was ported from Lean 3 source module category_theory.glue_data
! leanprover-community/mathlib commit 14b69e9f3c16630440a2cbd46f1ddad0d561dee7
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Tactic.Elementwise
import Mathbin.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathbin.CategoryTheory.Limits.Constructions.EpiMono
import Mathbin.CategoryTheory.Limits.Preserves.Limits
import Mathbin.CategoryTheory.Limits.Shapes.Types

/-!
# Gluing data

We define `glue_data` as a family of data needed to glue topological spaces, schemes, etc. We
provide the API to realize it as a multispan diagram, and also states lemmas about its
interaction with a functor that preserves certain pullbacks.

-/

noncomputable section

open CategoryTheory.Limits

namespace CategoryTheory

universe v u₁ u₂

variable (C : Type u₁) [Category.{v} C] {C' : Type u₂} [Category.{v} C']

#print CategoryTheory.GlueData /-
/-- A gluing datum consists of
1. An index type `J`
2. An object `U i` for each `i : J`.
3. An object `V i j` for each `i j : J`.
4. A monomorphism `f i j : V i j ⟶ U i` for each `i j : J`.
5. A transition map `t i j : V i j ⟶ V j i` for each `i j : J`.
such that
6. `f i i` is an isomorphism.
7. `t i i` is the identity.
8. The pullback for `f i j` and `f i k` exists.
9. `V i j ×[U i] V i k ⟶ V i j ⟶ V j i` factors through `V j k ×[U j] V j i ⟶ V j i` via some
    `t' : V i j ×[U i] V i k ⟶ V j k ×[U j] V j i`.
10. `t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _`.
-/
@[nolint has_nonempty_instance]
structure GlueData where
  J : Type v
  U : J → C
  V : J × J → C
  f : ∀ i j, V (i, j) ⟶ U i
  f_mono : ∀ i j, Mono (f i j) := by infer_instance
  f_hasPullback : ∀ i j k, HasPullback (f i j) (f i k) := by infer_instance
  f_id : ∀ i, IsIso (f i i) := by infer_instance
  t : ∀ i j, V (i, j) ⟶ V (j, i)
  t_id : ∀ i, t i i = 𝟙 _
  t' : ∀ i j k, pullback (f i j) (f i k) ⟶ pullback (f j k) (f j i)
  t_fac : ∀ i j k, t' i j k ≫ pullback.snd = pullback.fst ≫ t i j
  cocycle : ∀ i j k, t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _
#align category_theory.glue_data CategoryTheory.GlueData
-/

attribute [simp] glue_data.t_id

attribute [instance] glue_data.f_id glue_data.f_mono glue_data.f_has_pullback

attribute [reassoc.1] glue_data.t_fac glue_data.cocycle

namespace GlueData

variable {C} (D : GlueData C)

#print CategoryTheory.GlueData.t'_iij /-
@[simp]
theorem t'_iij (i j : D.J) : D.t' i i j = (pullbackSymmetry _ _).Hom :=
  by
  have eq₁ := D.t_fac i i j
  have eq₂ := (is_iso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _)
  rw [D.t_id, category.comp_id, eq₂] at eq₁
  have eq₃ := (is_iso.eq_comp_inv (D.f i i)).mp eq₁
  rw [category.assoc, ← pullback.condition, ← category.assoc] at eq₃
  exact
    mono.right_cancellation _ _
      ((mono.right_cancellation _ _ eq₃).trans (pullback_symmetry_hom_comp_fst _ _).symm)
#align category_theory.glue_data.t'_iij CategoryTheory.GlueData.t'_iij
-/

#print CategoryTheory.GlueData.t'_jii /-
theorem t'_jii (i j : D.J) : D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd :=
  by
  rw [← category.assoc, ← D.t_fac]
  simp
#align category_theory.glue_data.t'_jii CategoryTheory.GlueData.t'_jii
-/

#print CategoryTheory.GlueData.t'_iji /-
theorem t'_iji (i j : D.J) : D.t' i j i = pullback.fst ≫ D.t i j ≫ inv pullback.snd :=
  by
  rw [← category.assoc, ← D.t_fac]
  simp
#align category_theory.glue_data.t'_iji CategoryTheory.GlueData.t'_iji
-/

#print CategoryTheory.GlueData.t_inv /-
@[simp, reassoc.1, elementwise]
theorem t_inv (i j : D.J) : D.t i j ≫ D.t j i = 𝟙 _ :=
  by
  have eq : (pullback_symmetry (D.f i i) (D.f i j)).Hom = pullback.snd ≫ inv pullback.fst := by simp
  have := D.cocycle i j i
  rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, Eq] at this
  simp only [category.assoc, is_iso.inv_hom_id_assoc] at this
  rw [← is_iso.eq_inv_comp, ← category.assoc, is_iso.comp_inv_eq] at this
  simpa using this
#align category_theory.glue_data.t_inv CategoryTheory.GlueData.t_inv
-/

#print CategoryTheory.GlueData.t'_inv /-
theorem t'_inv (i j k : D.J) :
    D.t' i j k ≫ (pullbackSymmetry _ _).Hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).Hom = 𝟙 _ :=
  by
  rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)]
  simp [t_fac, t_fac_assoc]
#align category_theory.glue_data.t'_inv CategoryTheory.GlueData.t'_inv
-/

#print CategoryTheory.GlueData.t_isIso /-
instance t_isIso (i j : D.J) : IsIso (D.t i j) :=
  ⟨⟨D.t j i, D.t_inv _ _, D.t_inv _ _⟩⟩
#align category_theory.glue_data.t_is_iso CategoryTheory.GlueData.t_isIso
-/

#print CategoryTheory.GlueData.t'_isIso /-
instance t'_isIso (i j k : D.J) : IsIso (D.t' i j k) :=
  ⟨⟨D.t' j k i ≫ D.t' k i j, D.cocycle _ _ _, by simpa using D.cocycle _ _ _⟩⟩
#align category_theory.glue_data.t'_is_iso CategoryTheory.GlueData.t'_isIso
-/

#print CategoryTheory.GlueData.t'_comp_eq_pullbackSymmetry /-
@[reassoc.1]
theorem t'_comp_eq_pullbackSymmetry (i j k : D.J) :
    D.t' j k i ≫ D.t' k i j =
      (pullbackSymmetry _ _).Hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).Hom :=
  by
  trans inv (D.t' i j k)
  · exact is_iso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)
  · rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)]
    simp [t_fac, t_fac_assoc]
#align category_theory.glue_data.t'_comp_eq_pullback_symmetry CategoryTheory.GlueData.t'_comp_eq_pullbackSymmetry
-/

#print CategoryTheory.GlueData.sigmaOpens /-
/-- (Implementation) The disjoint union of `U i`. -/
def sigmaOpens [HasCoproduct D.U] : C :=
  ∐ D.U
#align category_theory.glue_data.sigma_opens CategoryTheory.GlueData.sigmaOpens
-/

#print CategoryTheory.GlueData.diagram /-
/-- (Implementation) The diagram to take colimit of. -/
def diagram : MultispanIndex C where
  L := D.J × D.J
  R := D.J
  fstFrom := Prod.fst
  sndFrom := Prod.snd
  left := D.V
  right := D.U
  fst := fun ⟨i, j⟩ => D.f i j
  snd := fun ⟨i, j⟩ => D.t i j ≫ D.f j i
#align category_theory.glue_data.diagram CategoryTheory.GlueData.diagram
-/

#print CategoryTheory.GlueData.diagram_l /-
@[simp]
theorem diagram_l : D.diagram.L = (D.J × D.J) :=
  rfl
#align category_theory.glue_data.diagram_L CategoryTheory.GlueData.diagram_l
-/

#print CategoryTheory.GlueData.diagram_r /-
@[simp]
theorem diagram_r : D.diagram.R = D.J :=
  rfl
#align category_theory.glue_data.diagram_R CategoryTheory.GlueData.diagram_r
-/

#print CategoryTheory.GlueData.diagram_fstFrom /-
@[simp]
theorem diagram_fstFrom (i j : D.J) : D.diagram.fstFrom ⟨i, j⟩ = i :=
  rfl
#align category_theory.glue_data.diagram_fst_from CategoryTheory.GlueData.diagram_fstFrom
-/

#print CategoryTheory.GlueData.diagram_sndFrom /-
@[simp]
theorem diagram_sndFrom (i j : D.J) : D.diagram.sndFrom ⟨i, j⟩ = j :=
  rfl
#align category_theory.glue_data.diagram_snd_from CategoryTheory.GlueData.diagram_sndFrom
-/

#print CategoryTheory.GlueData.diagram_fst /-
@[simp]
theorem diagram_fst (i j : D.J) : D.diagram.fst ⟨i, j⟩ = D.f i j :=
  rfl
#align category_theory.glue_data.diagram_fst CategoryTheory.GlueData.diagram_fst
-/

#print CategoryTheory.GlueData.diagram_snd /-
@[simp]
theorem diagram_snd (i j : D.J) : D.diagram.snd ⟨i, j⟩ = D.t i j ≫ D.f j i :=
  rfl
#align category_theory.glue_data.diagram_snd CategoryTheory.GlueData.diagram_snd
-/

#print CategoryTheory.GlueData.diagram_left /-
@[simp]
theorem diagram_left : D.diagram.left = D.V :=
  rfl
#align category_theory.glue_data.diagram_left CategoryTheory.GlueData.diagram_left
-/

#print CategoryTheory.GlueData.diagram_right /-
@[simp]
theorem diagram_right : D.diagram.right = D.U :=
  rfl
#align category_theory.glue_data.diagram_right CategoryTheory.GlueData.diagram_right
-/

section

variable [HasMulticoequalizer D.diagram]

#print CategoryTheory.GlueData.glued /-
/-- The glued object given a family of gluing data. -/
def glued : C :=
  multicoequalizer D.diagram
#align category_theory.glue_data.glued CategoryTheory.GlueData.glued
-/

#print CategoryTheory.GlueData.ι /-
/-- The map `D.U i ⟶ D.glued` for each `i`. -/
def ι (i : D.J) : D.U i ⟶ D.glued :=
  Multicoequalizer.π D.diagram i
#align category_theory.glue_data.ι CategoryTheory.GlueData.ι
-/

#print CategoryTheory.GlueData.glue_condition /-
@[simp, elementwise]
theorem glue_condition (i j : D.J) : D.t i j ≫ D.f j i ≫ D.ι j = D.f i j ≫ D.ι i :=
  (Category.assoc _ _ _).symm.trans (Multicoequalizer.condition D.diagram ⟨i, j⟩).symm
#align category_theory.glue_data.glue_condition CategoryTheory.GlueData.glue_condition
-/

#print CategoryTheory.GlueData.vPullbackCone /-
/-- The pullback cone spanned by `V i j ⟶ U i` and `V i j ⟶ U j`.
This will often be a pullback diagram. -/
def vPullbackCone (i j : D.J) : PullbackCone (D.ι i) (D.ι j) :=
  PullbackCone.mk (D.f i j) (D.t i j ≫ D.f j i) (by simp)
#align category_theory.glue_data.V_pullback_cone CategoryTheory.GlueData.vPullbackCone
-/

variable [HasColimits C]

#print CategoryTheory.GlueData.π /-
/-- The projection `∐ D.U ⟶ D.glued` given by the colimit. -/
def π : D.sigmaOpens ⟶ D.glued :=
  Multicoequalizer.sigmaπ D.diagram
#align category_theory.glue_data.π CategoryTheory.GlueData.π
-/

#print CategoryTheory.GlueData.π_epi /-
instance π_epi : Epi D.π := by
  unfold π
  infer_instance
#align category_theory.glue_data.π_epi CategoryTheory.GlueData.π_epi
-/

end

#print CategoryTheory.GlueData.types_π_surjective /-
theorem types_π_surjective (D : GlueData (Type _)) : Function.Surjective D.π :=
  (epi_iff_surjective _).mp inferInstance
#align category_theory.glue_data.types_π_surjective CategoryTheory.GlueData.types_π_surjective
-/

#print CategoryTheory.GlueData.types_ι_jointly_surjective /-
theorem types_ι_jointly_surjective (D : GlueData (Type _)) (x : D.glued) :
    ∃ (i : _)(y : D.U i), D.ι i y = x :=
  by
  delta CategoryTheory.GlueData.ι
  simp_rw [← multicoequalizer.ι_sigma_π D.diagram]
  rcases D.types_π_surjective x with ⟨x', rfl⟩
  have := colimit.iso_colimit_cocone (types.coproduct_colimit_cocone _)
  rw [←
    show (colimit.iso_colimit_cocone (types.coproduct_colimit_cocone _)).inv _ = x' from
      concrete_category.congr_hom
        (colimit.iso_colimit_cocone (types.coproduct_colimit_cocone _)).hom_inv_id x']
  rcases(colimit.iso_colimit_cocone (types.coproduct_colimit_cocone _)).Hom x' with ⟨i, y⟩
  exact ⟨i, y, by simpa [← multicoequalizer.ι_sigma_π, -multicoequalizer.ι_sigma_π] ⟩
#align category_theory.glue_data.types_ι_jointly_surjective CategoryTheory.GlueData.types_ι_jointly_surjective
-/

variable (F : C ⥤ C') [H : ∀ i j k, PreservesLimit (cospan (D.f i j) (D.f i k)) F]

include H

instance (i j k : D.J) : HasPullback (F.map (D.f i j)) (F.map (D.f i k)) :=
  ⟨⟨⟨_, isLimitOfHasPullbackOfPreservesLimit F (D.f i j) (D.f i k)⟩⟩⟩

#print CategoryTheory.GlueData.mapGlueData /-
/-- A functor that preserves the pullbacks of `f i j` and `f i k` can map a family of glue data. -/
@[simps]
def mapGlueData : GlueData C' where
  J := D.J
  U i := F.obj (D.U i)
  V i := F.obj (D.V i)
  f i j := F.map (D.f i j)
  f_mono i j := preserves_mono_of_preservesLimit _ _
  f_id i := inferInstance
  t i j := F.map (D.t i j)
  t_id i := by
    rw [D.t_id i]
    simp
  t' i j k :=
    (PreservesPullback.iso F (D.f i j) (D.f i k)).inv ≫
      F.map (D.t' i j k) ≫ (PreservesPullback.iso F (D.f j k) (D.f j i)).Hom
  t_fac i j k := by simpa [iso.inv_comp_eq] using congr_arg (fun f => F.map f) (D.t_fac i j k)
  cocycle i j k := by
    simp only [category.assoc, iso.hom_inv_id_assoc, ← functor.map_comp_assoc, D.cocycle,
      iso.inv_hom_id, CategoryTheory.Functor.map_id, category.id_comp]
#align category_theory.glue_data.map_glue_data CategoryTheory.GlueData.mapGlueData
-/

#print CategoryTheory.GlueData.diagramIso /-
/-- The diagram of the image of a `glue_data` under a functor `F` is naturally isomorphic to the
original diagram of the `glue_data` via `F`.
-/
def diagramIso : D.diagram.multispan ⋙ F ≅ (D.mapGlueData F).diagram.multispan :=
  NatIso.ofComponents
    (fun x =>
      match x with
      | walking_multispan.left a => Iso.refl _
      | walking_multispan.right b => Iso.refl _)
    (by
      rintro (⟨_, _⟩ | _) _ (_ | _ | _)
      · erw [category.comp_id, category.id_comp, Functor.map_id]
        rfl
      · erw [category.comp_id, category.id_comp]
        rfl
      · erw [category.comp_id, category.id_comp, functor.map_comp]
        rfl
      · erw [category.comp_id, category.id_comp, Functor.map_id]
        rfl)
#align category_theory.glue_data.diagram_iso CategoryTheory.GlueData.diagramIso
-/

/- warning: category_theory.glue_data.diagram_iso_app_left -> CategoryTheory.GlueData.diagramIso_app_left is a dubious translation:
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C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) 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(CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.GlueData.diagramIso.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (CategoryTheory.Iso.refl.{u1, u3} C' _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {C' : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C'] (D : CategoryTheory.GlueData.{u1, u2} C _inst_1) (F : CategoryTheory.Functor.{u1, u1, u2, u3} C _inst_1 C' _inst_2) [H : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, u3} C _inst_1 C' _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : Prod.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D)), Eq.{succ u1} (CategoryTheory.Iso.{u1, u3} C' _inst_2 (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F)) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} 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(CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i))) (CategoryTheory.Iso.app.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) 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u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) 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u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F)) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.diagram_iso_app_left CategoryTheory.GlueData.diagramIso_app_leftₓ'. -/
@[simp]
theorem diagramIso_app_left (i : D.J × D.J) :
    (D.diagramIso F).app (WalkingMultispan.left i) = Iso.refl _ :=
  rfl
#align category_theory.glue_data.diagram_iso_app_left CategoryTheory.GlueData.diagramIso_app_left

/- warning: category_theory.glue_data.diagram_iso_app_right -> CategoryTheory.GlueData.diagramIso_app_right is a dubious translation:
lean 3 declaration is
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C _inst_1 D) i k)) (CategoryTheory.GlueData.u.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), Eq.{succ u1} (CategoryTheory.Iso.{u1, u3} C' _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i))) (CategoryTheory.Iso.app.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.GlueData.diagramIso.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (CategoryTheory.Iso.refl.{u1, u3} C' _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {C' : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C'] (D : CategoryTheory.GlueData.{u1, u2} C _inst_1) (F : CategoryTheory.Functor.{u1, u1, u2, u3} C _inst_1 C' _inst_2) [H : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, u3} C _inst_1 C' _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), Eq.{succ u1} (CategoryTheory.Iso.{u1, u3} C' _inst_2 (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} 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Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.diagram_iso_app_right CategoryTheory.GlueData.diagramIso_app_rightₓ'. -/
@[simp]
theorem diagramIso_app_right (i : D.J) :
    (D.diagramIso F).app (WalkingMultispan.right i) = Iso.refl _ :=
  rfl
#align category_theory.glue_data.diagram_iso_app_right CategoryTheory.GlueData.diagramIso_app_right

/- warning: category_theory.glue_data.diagram_iso_hom_app_left -> CategoryTheory.GlueData.diagramIso_hom_app_left is a dubious translation:
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_inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (CategoryTheory.CategoryStruct.id.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {C' : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C'] (D : CategoryTheory.GlueData.{u1, u2} C _inst_1) (F : CategoryTheory.Functor.{u1, u1, u2, u3} C _inst_1 C' _inst_2) [H : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, u3} C _inst_1 C' _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : Prod.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D)), Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} 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(CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F)) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) 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(CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) 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i)))
Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.diagram_iso_hom_app_left CategoryTheory.GlueData.diagramIso_hom_app_leftₓ'. -/
@[simp]
theorem diagramIso_hom_app_left (i : D.J × D.J) :
    (D.diagramIso F).Hom.app (WalkingMultispan.left i) = 𝟙 _ :=
  rfl
#align category_theory.glue_data.diagram_iso_hom_app_left CategoryTheory.GlueData.diagramIso_hom_app_left

/- warning: category_theory.glue_data.diagram_iso_hom_app_right -> CategoryTheory.GlueData.diagramIso_hom_app_right is a dubious translation:
lean 3 declaration is
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C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {C' : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C'] (D : CategoryTheory.GlueData.{u1, u2} C _inst_1) (F : CategoryTheory.Functor.{u1, u1, u2, u3} C _inst_1 C' _inst_2) [H : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, u3} C _inst_1 C' _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F)) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} 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(CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} 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(CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i))) (CategoryTheory.NatTrans.app.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C _inst_1 C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.Iso.hom.{u1, max u3 u1} (CategoryTheory.Functor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 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Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.diagram_iso_hom_app_right CategoryTheory.GlueData.diagramIso_hom_app_rightₓ'. -/
@[simp]
theorem diagramIso_hom_app_right (i : D.J) :
    (D.diagramIso F).Hom.app (WalkingMultispan.right i) = 𝟙 _ :=
  rfl
#align category_theory.glue_data.diagram_iso_hom_app_right CategoryTheory.GlueData.diagramIso_hom_app_right

/- warning: category_theory.glue_data.diagram_iso_inv_app_left -> CategoryTheory.GlueData.diagramIso_inv_app_left is a dubious translation:
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C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.GlueData.diagramIso.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k))) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (CategoryTheory.CategoryStruct.id.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.Limits.WalkingMultispan.left.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {C' : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C'] (D : CategoryTheory.GlueData.{u1, u2} C _inst_1) (F : CategoryTheory.Functor.{u1, u1, u2, u3} C _inst_1 C' _inst_2) [H : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, u3} C _inst_1 C' _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : Prod.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D)), Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} 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Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.diagram_iso_inv_app_left CategoryTheory.GlueData.diagramIso_inv_app_leftₓ'. -/
@[simp]
theorem diagramIso_inv_app_left (i : D.J × D.J) :
    (D.diagramIso F).inv.app (WalkingMultispan.left i) = 𝟙 _ :=
  rfl
#align category_theory.glue_data.diagram_iso_inv_app_left CategoryTheory.GlueData.diagramIso_inv_app_left

/- warning: category_theory.glue_data.diagram_iso_inv_app_right -> CategoryTheory.GlueData.diagramIso_inv_app_right is a dubious translation:
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C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F) (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.GlueData.diagramIso.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k))) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (CategoryTheory.CategoryStruct.id.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)))) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {C' : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C'] (D : CategoryTheory.GlueData.{u1, u2} C _inst_1) (F : CategoryTheory.Functor.{u1, u1, u2, u3} C _inst_1 C' _inst_2) [H : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, u3} C _inst_1 C' _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u3, u1} C' _inst_2 (CategoryTheory.GlueData.diagram.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k))))) (CategoryTheory.Limits.WalkingMultispan.right.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) i)) (Prefunctor.obj.{succ u1, succ u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))))) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) C' _inst_2 (CategoryTheory.Functor.comp.{u1, u1, u1, u1, u2, u3} (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 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Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.diagram_iso_inv_app_right CategoryTheory.GlueData.diagramIso_inv_app_rightₓ'. -/
@[simp]
theorem diagramIso_inv_app_right (i : D.J) :
    (D.diagramIso F).inv.app (WalkingMultispan.right i) = 𝟙 _ :=
  rfl
#align category_theory.glue_data.diagram_iso_inv_app_right CategoryTheory.GlueData.diagramIso_inv_app_right

variable [HasMulticoequalizer D.diagram] [PreservesColimit D.diagram.multispan F]

omit H

#print CategoryTheory.GlueData.hasColimit_multispan_comp /-
theorem hasColimit_multispan_comp : HasColimit (D.diagram.multispan ⋙ F) :=
  ⟨⟨⟨_, PreservesColimit.preserves (colimit.isColimit _)⟩⟩⟩
#align category_theory.glue_data.has_colimit_multispan_comp CategoryTheory.GlueData.hasColimit_multispan_comp
-/

include H

attribute [local instance] has_colimit_multispan_comp

#print CategoryTheory.GlueData.hasColimit_mapGlueData_diagram /-
theorem hasColimit_mapGlueData_diagram : HasMulticoequalizer (D.mapGlueData F).diagram :=
  hasColimitOfIso (D.diagramIso F).symm
#align category_theory.glue_data.has_colimit_map_glue_data_diagram CategoryTheory.GlueData.hasColimit_mapGlueData_diagram
-/

attribute [local instance] has_colimit_map_glue_data_diagram

/- warning: category_theory.glue_data.glued_iso -> CategoryTheory.GlueData.gluedIso is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.glued_iso CategoryTheory.GlueData.gluedIsoₓ'. -/
/-- If `F` preserves the gluing, we obtain an iso between the glued objects. -/
def gluedIso : F.obj D.glued ≅ (D.mapGlueData F).glued :=
  preservesColimitIso F D.diagram.multispan ≪≫ Limits.HasColimit.isoOfNatIso (D.diagramIso F)
#align category_theory.glue_data.glued_iso CategoryTheory.GlueData.gluedIso

/- warning: category_theory.glue_data.ι_glued_iso_hom -> CategoryTheory.GlueData.ι_gluedIso_hom is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.ι_glued_iso_hom CategoryTheory.GlueData.ι_gluedIso_homₓ'. -/
@[simp, reassoc.1]
theorem ι_gluedIso_hom (i : D.J) : F.map (D.ι i) ≫ (D.gluedIso F).Hom = (D.mapGlueData F).ι i :=
  by
  erw [ι_preserves_colimits_iso_hom_assoc]
  rw [has_colimit.iso_of_nat_iso_ι_hom]
  erw [category.id_comp]
  rfl
#align category_theory.glue_data.ι_glued_iso_hom CategoryTheory.GlueData.ι_gluedIso_hom

/- warning: category_theory.glue_data.ι_glued_iso_inv -> CategoryTheory.GlueData.ι_gluedIso_inv is a dubious translation:
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C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] [_inst_3 : CategoryTheory.Limits.HasMulticoequalizer.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)] [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, u3} C _inst_1 C' _inst_2 (CategoryTheory.Limits.WalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) 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u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)) i) (Prefunctor.obj.{succ u1, succ u1, u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u3} C _inst_1 C' _inst_2 F) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3))) (CategoryTheory.CategoryStruct.comp.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2) (CategoryTheory.GlueData.U.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : 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_inst_2 F) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3)) (CategoryTheory.GlueData.ι.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)) (CategoryTheory.GlueData.hasColimit_mapGlueData_diagram.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k) _inst_3 _inst_4) i) (CategoryTheory.Iso.inv.{u1, u3} C' _inst_2 (Prefunctor.obj.{succ u1, succ u1, u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u3} C _inst_1 C' _inst_2 F) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3)) (CategoryTheory.GlueData.glued.{u1, u3} C' _inst_2 (CategoryTheory.GlueData.mapGlueData.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k)) (CategoryTheory.GlueData.hasColimit_mapGlueData_diagram.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k) _inst_3 _inst_4)) (CategoryTheory.GlueData.gluedIso.{u1, u2, u3} C _inst_1 C' _inst_2 D F (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => H i j k) _inst_3 _inst_4))) (Prefunctor.map.{succ u1, succ u1, u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) C' (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C' (CategoryTheory.Category.toCategoryStruct.{u1, u3} C' _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u3} C _inst_1 C' _inst_2 F) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3) (CategoryTheory.GlueData.ι.{u1, u2} C _inst_1 D _inst_3 i))
Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.ι_glued_iso_inv CategoryTheory.GlueData.ι_gluedIso_invₓ'. -/
@[simp, reassoc.1]
theorem ι_gluedIso_inv (i : D.J) : (D.mapGlueData F).ι i ≫ (D.gluedIso F).inv = F.map (D.ι i) := by
  rw [iso.comp_inv_eq, ι_glued_iso_hom]
#align category_theory.glue_data.ι_glued_iso_inv CategoryTheory.GlueData.ι_gluedIso_inv

#print CategoryTheory.GlueData.vPullbackConeIsLimitOfMap /-
/-- If `F` preserves the gluing, and reflects the pullback of `U i ⟶ glued` and `U j ⟶ glued`,
then `F` reflects the fact that `V_pullback_cone` is a pullback. -/
def vPullbackConeIsLimitOfMap (i j : D.J) [ReflectsLimit (cospan (D.ι i) (D.ι j)) F]
    (hc : IsLimit ((D.mapGlueData F).vPullbackCone i j)) : IsLimit (D.vPullbackCone i j) :=
  by
  apply is_limit_of_reflects F
  apply (is_limit_map_cone_pullback_cone_equiv _ _).symm _
  let e :
    cospan (F.map (D.ι i)) (F.map (D.ι j)) ≅
      cospan ((D.map_glue_data F).ι i) ((D.map_glue_data F).ι j)
  exact
    nat_iso.of_components
      (fun x => by
        cases x
        exacts[D.glued_iso F, iso.refl _])
      (by rintro (_ | _) (_ | _) (_ | _ | _) <;> simp)
  apply is_limit.postcompose_hom_equiv e _ _
  apply hc.of_iso_limit
  refine' cones.ext (iso.refl _) _
  · rintro (_ | _ | _)
    change _ = _ ≫ (_ ≫ _) ≫ _
    all_goals change _ = 𝟙 _ ≫ _ ≫ _; simpa
#align category_theory.glue_data.V_pullback_cone_is_limit_of_map CategoryTheory.GlueData.vPullbackConeIsLimitOfMap
-/

omit H

/- warning: category_theory.glue_data.ι_jointly_surjective -> CategoryTheory.GlueData.ι_jointly_surjective is a dubious translation:
lean 3 declaration is
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but is expected to have type
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_inst_1 D))) (CategoryTheory.Limits.WalkingMultispan.instSmallCategoryWalkingMultispan.{u1} (CategoryTheory.Limits.MultispanIndex.L.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.R.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.fstFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) (CategoryTheory.Limits.MultispanIndex.sndFrom.{u1, u1, u2} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D))) (CategoryTheory.Limits.MultispanIndex.multispan.{u1, u2, u1} C _inst_1 (CategoryTheory.GlueData.diagram.{u1, u2} C _inst_1 D)) F] [_inst_6 : forall (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (j : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (k : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u2} C _inst_1 (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i j)) (CategoryTheory.GlueData.V.{u1, u2} C _inst_1 D (Prod.mk.{u1, u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) i k)) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i j) (CategoryTheory.GlueData.f.{u1, u2} C _inst_1 D i k)) F] (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} F) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3)), Exists.{succ u1} (CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) (fun (i : CategoryTheory.GlueData.J.{u1, u2} C _inst_1 D) => Exists.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} F) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i)) (fun (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} F) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i)) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} F) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} F) (CategoryTheory.GlueData.U.{u1, u2} C _inst_1 D i) (CategoryTheory.GlueData.glued.{u1, u2} C _inst_1 D _inst_3) (CategoryTheory.GlueData.ι.{u1, u2} C _inst_1 D _inst_3 i) y) x))
Case conversion may be inaccurate. Consider using '#align category_theory.glue_data.ι_jointly_surjective CategoryTheory.GlueData.ι_jointly_surjectiveₓ'. -/
/-- If there is a forgetful functor into `Type` that preserves enough (co)limits, then `D.ι` will
be jointly surjective. -/
theorem ι_jointly_surjective (F : C ⥤ Type v) [PreservesColimit D.diagram.multispan F]
    [∀ i j k : D.J, PreservesLimit (cospan (D.f i j) (D.f i k)) F] (x : F.obj D.glued) :
    ∃ (i : _)(y : F.obj (D.U i)), F.map (D.ι i) y = x :=
  by
  let e := D.glued_iso F
  obtain ⟨i, y, eq⟩ := (D.map_glue_data F).types_ι_jointly_surjective (e.hom x)
  replace eq := congr_arg e.inv Eq
  change ((D.map_glue_data F).ι i ≫ e.inv) y = (e.hom ≫ e.inv) x at eq
  rw [e.hom_inv_id, D.ι_glued_iso_inv] at eq
  exact ⟨i, y, Eq⟩
#align category_theory.glue_data.ι_jointly_surjective CategoryTheory.GlueData.ι_jointly_surjective

end GlueData

end CategoryTheory