/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison

! This file was ported from Lean 3 source module category_theory.limits.functor_category
! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Preserves.Limits

/-!
# (Co)limits in functor categories.

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

We show that if `D` has limits, then the functor category `C ⥤ D` also has limits
(`category_theory.limits.functor_category_has_limits`),
and the evaluation functors preserve limits
(`category_theory.limits.evaluation_preserves_limits`)
(and similarly for colimits).

We also show that `F : D ⥤ K ⥤ C` preserves (co)limits if it does so for each `k : K`
(`category_theory.limits.preserves_limits_of_evaluation` and
`category_theory.limits.preserves_colimits_of_evaluation`).
-/

open CategoryTheory CategoryTheory.Category CategoryTheory.Functor

-- morphism levels before object levels. See note [category_theory universes].
universe w' w v₁ v₂ u₁ u₂ v v' u u'

namespace CategoryTheory.Limits

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

variable {J : Type u₁} [Category.{v₁} J] {K : Type u₂} [Category.{v₂} K]

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit.lift_π_app CategoryTheory.Limits.limit.lift_π_appₓ'. -/
@[simp, reassoc.1]
theorem limit.lift_π_app (H : J ⥤ K ⥤ C) [HasLimit H] (c : Cone H) (j : J) (k : K) :
    (limit.lift H c).app k ≫ (limit.π H j).app k = (c.π.app j).app k :=
  congr_app (limit.lift_π c j) k
#align category_theory.limits.limit.lift_π_app CategoryTheory.Limits.limit.lift_π_app

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but is expected to have type
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u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max u4 u5, max u3 u4 u5, max (max (max u6 u4) u5) u2, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) (CategoryTheory.Limits.Cocone.ι.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) j) k)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit.ι_desc_app CategoryTheory.Limits.colimit.ι_desc_appₓ'. -/
@[simp, reassoc.1]
theorem colimit.ι_desc_app (H : J ⥤ K ⥤ C) [HasColimit H] (c : Cocone H) (j : J) (k : K) :
    (colimit.ι H j).app k ≫ (colimit.desc H c).app k = (c.ι.app j).app k :=
  congr_app (colimit.ι_desc c j) k
#align category_theory.limits.colimit.ι_desc_app CategoryTheory.Limits.colimit.ι_desc_app

/- warning: category_theory.limits.evaluation_jointly_reflects_limits -> CategoryTheory.Limits.evaluationJointlyReflectsLimits is a dubious translation:
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  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} (c : CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F), (forall (k : K), CategoryTheory.Limits.IsLimit.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, 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Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_jointly_reflects_limits CategoryTheory.Limits.evaluationJointlyReflectsLimitsₓ'. -/
/-- The evaluation functors jointly reflect limits: that is, to show a cone is a limit of `F`
it suffices to show that each evaluation cone is a limit. In other words, to prove a cone is
limiting you can show it's pointwise limiting.
-/
def evaluationJointlyReflectsLimits {F : J ⥤ K ⥤ C} (c : Cone F)
    (t : ∀ k : K, IsLimit (((evaluation K C).obj k).mapCone c)) : IsLimit c
    where
  lift s :=
    { app := fun k => (t k).lift ⟨s.pt.obj k, whiskerRight s.π ((evaluation K C).obj k)⟩
      naturality' := fun X Y f =>
        (t Y).hom_ext fun j => by
          rw [assoc, (t Y).fac _ j]
          simpa using
            ((t X).fac_assoc ⟨s.X.obj X, whisker_right s.π ((evaluation K C).obj X)⟩ j _).symm }
  fac s j := NatTrans.ext _ _ <| funext fun k => (t k).fac _ j
  uniq s m w :=
    NatTrans.ext _ _ <|
      funext fun x =>
        (t x).hom_ext fun j =>
          (congr_app (w j) x).trans
            ((t x).fac ⟨s.pt.obj _, whiskerRight s.π ((evaluation K C).obj _)⟩ j).symm
#align category_theory.limits.evaluation_jointly_reflects_limits CategoryTheory.Limits.evaluationJointlyReflectsLimits

/- warning: category_theory.limits.combine_cones -> CategoryTheory.Limits.combineCones is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.combine_cones CategoryTheory.Limits.combineConesₓ'. -/
/-- Given a functor `F` and a collection of limit cones for each diagram `X ↦ F X k`, we can stitch
them together to give a cone for the diagram `F`.
`combined_is_limit` shows that the new cone is limiting, and `eval_combined` shows it is
(essentially) made up of the original cones.
-/
@[simps]
def combineCones (F : J ⥤ K ⥤ C) (c : ∀ k : K, LimitCone (F.flip.obj k)) : Cone F
    where
  pt :=
    { obj := fun k => (c k).Cone.pt
      map := fun k₁ k₂ f => (c k₂).IsLimit.lift ⟨_, (c k₁).Cone.π ≫ F.flip.map f⟩
      map_id' := fun k =>
        (c k).IsLimit.hom_ext fun j => by
          dsimp
          simp
      map_comp' := fun k₁ k₂ k₃ f₁ f₂ => (c k₃).IsLimit.hom_ext fun j => by simp }
  π :=
    { app := fun j => { app := fun k => (c k).Cone.π.app j }
      naturality' := fun j₁ j₂ g => NatTrans.ext _ _ <| funext fun k => (c k).Cone.π.naturality g }
#align category_theory.limits.combine_cones CategoryTheory.Limits.combineCones

/- warning: category_theory.limits.evaluate_combined_cones -> CategoryTheory.Limits.evaluateCombinedCones is a dubious translation:
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C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k) (c k))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluate_combined_cones CategoryTheory.Limits.evaluateCombinedConesₓ'. -/
/-- The stitched together cones each project down to the original given cones (up to iso). -/
def evaluateCombinedCones (F : J ⥤ K ⥤ C) (c : ∀ k : K, LimitCone (F.flip.obj k)) (k : K) :
    ((evaluation K C).obj k).mapCone (combineCones F c) ≅ (c k).Cone :=
  Cones.ext (Iso.refl _) (by tidy)
#align category_theory.limits.evaluate_combined_cones CategoryTheory.Limits.evaluateCombinedCones

/- warning: category_theory.limits.combined_is_limit -> CategoryTheory.Limits.combinedIsLimit is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.combined_is_limit CategoryTheory.Limits.combinedIsLimitₓ'. -/
/-- Stitching together limiting cones gives a limiting cone. -/
def combinedIsLimit (F : J ⥤ K ⥤ C) (c : ∀ k : K, LimitCone (F.flip.obj k)) :
    IsLimit (combineCones F c) :=
  evaluationJointlyReflectsLimits _ fun k =>
    (c k).IsLimit.ofIsoLimit (evaluateCombinedCones F c k).symm
#align category_theory.limits.combined_is_limit CategoryTheory.Limits.combinedIsLimit

/- warning: category_theory.limits.evaluation_jointly_reflects_colimits -> CategoryTheory.Limits.evaluationJointlyReflectsColimits is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_jointly_reflects_colimits CategoryTheory.Limits.evaluationJointlyReflectsColimitsₓ'. -/
/-- The evaluation functors jointly reflect colimits: that is, to show a cocone is a colimit of `F`
it suffices to show that each evaluation cocone is a colimit. In other words, to prove a cocone is
colimiting you can show it's pointwise colimiting.
-/
def evaluationJointlyReflectsColimits {F : J ⥤ K ⥤ C} (c : Cocone F)
    (t : ∀ k : K, IsColimit (((evaluation K C).obj k).mapCocone c)) : IsColimit c
    where
  desc s :=
    { app := fun k => (t k).desc ⟨s.pt.obj k, whiskerRight s.ι ((evaluation K C).obj k)⟩
      naturality' := fun X Y f =>
        (t X).hom_ext fun j => by
          rw [(t X).fac_assoc _ j]
          erw [← (c.ι.app j).naturality_assoc f]
          erw [(t Y).fac ⟨s.X.obj _, whisker_right s.ι _⟩ j]
          dsimp
          simp }
  fac s j := NatTrans.ext _ _ <| funext fun k => (t k).fac _ j
  uniq s m w :=
    NatTrans.ext _ _ <|
      funext fun x =>
        (t x).hom_ext fun j =>
          (congr_app (w j) x).trans
            ((t x).fac ⟨s.pt.obj _, whiskerRight s.ι ((evaluation K C).obj _)⟩ j).symm
#align category_theory.limits.evaluation_jointly_reflects_colimits CategoryTheory.Limits.evaluationJointlyReflectsColimits

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.combine_cocones CategoryTheory.Limits.combineCoconesₓ'. -/
/--
Given a functor `F` and a collection of colimit cocones for each diagram `X ↦ F X k`, we can stitch
them together to give a cocone for the diagram `F`.
`combined_is_colimit` shows that the new cocone is colimiting, and `eval_combined` shows it is
(essentially) made up of the original cocones.
-/
@[simps]
def combineCocones (F : J ⥤ K ⥤ C) (c : ∀ k : K, ColimitCocone (F.flip.obj k)) : Cocone F
    where
  pt :=
    { obj := fun k => (c k).Cocone.pt
      map := fun k₁ k₂ f => (c k₁).IsColimit.desc ⟨_, F.flip.map f ≫ (c k₂).Cocone.ι⟩
      map_id' := fun k =>
        (c k).IsColimit.hom_ext fun j => by
          dsimp
          simp
      map_comp' := fun k₁ k₂ k₃ f₁ f₂ => (c k₁).IsColimit.hom_ext fun j => by simp }
  ι :=
    { app := fun j => { app := fun k => (c k).Cocone.ι.app j }
      naturality' := fun j₁ j₂ g =>
        NatTrans.ext _ _ <| funext fun k => (c k).Cocone.ι.naturality g }
#align category_theory.limits.combine_cocones CategoryTheory.Limits.combineCocones

/- warning: category_theory.limits.evaluate_combined_cocones -> CategoryTheory.Limits.evaluateCombinedCocones is a dubious translation:
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_inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k) (c k))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluate_combined_cocones CategoryTheory.Limits.evaluateCombinedCoconesₓ'. -/
/-- The stitched together cocones each project down to the original given cocones (up to iso). -/
def evaluateCombinedCocones (F : J ⥤ K ⥤ C) (c : ∀ k : K, ColimitCocone (F.flip.obj k)) (k : K) :
    ((evaluation K C).obj k).mapCocone (combineCocones F c) ≅ (c k).Cocone :=
  Cocones.ext (Iso.refl _) (by tidy)
#align category_theory.limits.evaluate_combined_cocones CategoryTheory.Limits.evaluateCombinedCocones

/- warning: category_theory.limits.combined_is_colimit -> CategoryTheory.Limits.combinedIsColimit is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.combined_is_colimit CategoryTheory.Limits.combinedIsColimitₓ'. -/
/-- Stitching together colimiting cocones gives a colimiting cocone. -/
def combinedIsColimit (F : J ⥤ K ⥤ C) (c : ∀ k : K, ColimitCocone (F.flip.obj k)) :
    IsColimit (combineCocones F c) :=
  evaluationJointlyReflectsColimits _ fun k =>
    (c k).IsColimit.ofIsoColimit (evaluateCombinedCocones F c k).symm
#align category_theory.limits.combined_is_colimit CategoryTheory.Limits.combinedIsColimit

noncomputable section

#print CategoryTheory.Limits.functorCategoryHasLimitsOfShape /-
instance functorCategoryHasLimitsOfShape [HasLimitsOfShape J C] : HasLimitsOfShape J (K ⥤ C)
    where HasLimit F :=
    HasLimit.mk
      { Cone := combineCones F fun k => getLimitCone _
        IsLimit := combinedIsLimit _ _ }
#align category_theory.limits.functor_category_has_limits_of_shape CategoryTheory.Limits.functorCategoryHasLimitsOfShape
-/

#print CategoryTheory.Limits.functorCategoryHasColimitsOfShape /-
instance functorCategoryHasColimitsOfShape [HasColimitsOfShape J C] : HasColimitsOfShape J (K ⥤ C)
    where HasColimit F :=
    HasColimit.mk
      { Cocone := combineCocones _ fun k => getColimitCocone _
        IsColimit := combinedIsColimit _ _ }
#align category_theory.limits.functor_category_has_colimits_of_shape CategoryTheory.Limits.functorCategoryHasColimitsOfShape
-/

#print CategoryTheory.Limits.functorCategoryHasLimitsOfSize /-
instance functorCategoryHasLimitsOfSize [HasLimitsOfSize.{v₁, u₁} C] :
    HasLimitsOfSize.{v₁, u₁} (K ⥤ C) :=
  ⟨inferInstance⟩
#align category_theory.limits.functor_category_has_limits_of_size CategoryTheory.Limits.functorCategoryHasLimitsOfSize
-/

#print CategoryTheory.Limits.functorCategoryHasColimitsOfSize /-
instance functorCategoryHasColimitsOfSize [HasColimitsOfSize.{v₁, u₁} C] :
    HasColimitsOfSize.{v₁, u₁} (K ⥤ C) :=
  ⟨inferInstance⟩
#align category_theory.limits.functor_category_has_colimits_of_size CategoryTheory.Limits.functorCategoryHasColimitsOfSize
-/

/- warning: category_theory.limits.evaluation_preserves_limits_of_shape -> CategoryTheory.Limits.evaluationPreservesLimitsOfShape is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_limits_of_shape CategoryTheory.Limits.evaluationPreservesLimitsOfShapeₓ'. -/
instance evaluationPreservesLimitsOfShape [HasLimitsOfShape J C] (k : K) :
    PreservesLimitsOfShape J ((evaluation K C).obj k)
    where PreservesLimit F :=
    preservesLimitOfPreservesLimitCone (combinedIsLimit _ _) <|
      IsLimit.ofIsoLimit (limit.isLimit _) (evaluateCombinedCones F _ k).symm
#align category_theory.limits.evaluation_preserves_limits_of_shape CategoryTheory.Limits.evaluationPreservesLimitsOfShape

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation CategoryTheory.Limits.limitObjIsoLimitCompEvaluationₓ'. -/
/-- If `F : J ⥤ K ⥤ C` is a functor into a functor category which has a limit,
then the evaluation of that limit at `k` is the limit of the evaluations of `F.obj j` at `k`.
-/
def limitObjIsoLimitCompEvaluation [HasLimitsOfShape J C] (F : J ⥤ K ⥤ C) (k : K) :
    (limit F).obj k ≅ limit (F ⋙ (evaluation K C).obj k) :=
  preservesLimitIso ((evaluation K C).obj k) F
#align category_theory.limits.limit_obj_iso_limit_comp_evaluation CategoryTheory.Limits.limitObjIsoLimitCompEvaluation

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_hom_π CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_πₓ'. -/
@[simp, reassoc.1]
theorem limitObjIsoLimitCompEvaluation_hom_π [HasLimitsOfShape J C] (F : J ⥤ K ⥤ C) (j : J)
    (k : K) :
    (limitObjIsoLimitCompEvaluation F k).Hom ≫ limit.π (F ⋙ (evaluation K C).obj k) j =
      (limit.π F j).app k :=
  by
  dsimp [limit_obj_iso_limit_comp_evaluation]
  simp
#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_hom_π CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_π

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_π_app CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_π_appₓ'. -/
@[simp, reassoc.1]
theorem limitObjIsoLimitCompEvaluation_inv_π_app [HasLimitsOfShape J C] (F : J ⥤ K ⥤ C) (j : J)
    (k : K) :
    (limitObjIsoLimitCompEvaluation F k).inv ≫ (limit.π F j).app k =
      limit.π (F ⋙ (evaluation K C).obj k) j :=
  by
  dsimp [limit_obj_iso_limit_comp_evaluation]
  rw [iso.inv_comp_eq]
  simp
#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_π_app CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_π_app

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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_map_limit_obj_iso_limit_comp_evaluation_hom CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_homₓ'. -/
@[simp, reassoc.1]
theorem limit_map_limitObjIsoLimitCompEvaluation_hom [HasLimitsOfShape J C] {i j : K}
    (F : J ⥤ K ⥤ C) (f : i ⟶ j) :
    (limit F).map f ≫ (limitObjIsoLimitCompEvaluation _ _).Hom =
      (limitObjIsoLimitCompEvaluation _ _).Hom ≫ limMap (whiskerLeft _ ((evaluation _ _).map f)) :=
  by
  ext
  dsimp
  simp
#align category_theory.limits.limit_map_limit_obj_iso_limit_comp_evaluation_hom CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom

/- warning: category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map -> CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_map is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_mapₓ'. -/
@[simp, reassoc.1]
theorem limitObjIsoLimitCompEvaluation_inv_limit_map [HasLimitsOfShape J C] {i j : K}
    (F : J ⥤ K ⥤ C) (f : i ⟶ j) :
    (limitObjIsoLimitCompEvaluation _ _).inv ≫ (limit F).map f =
      limMap (whiskerLeft _ ((evaluation _ _).map f)) ≫ (limitObjIsoLimitCompEvaluation _ _).inv :=
  by
  rw [iso.inv_comp_eq, ← category.assoc, iso.eq_comp_inv,
    limit_map_limit_obj_iso_limit_comp_evaluation_hom]
#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_map

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but is expected to have type
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H))) k)) f g)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_ext CategoryTheory.Limits.limit_obj_extₓ'. -/
@[ext]
theorem limit_obj_ext {H : J ⥤ K ⥤ C} [HasLimitsOfShape J C] {k : K} {W : C}
    {f g : W ⟶ (limit H).obj k}
    (w : ∀ j, f ≫ (Limits.limit.π H j).app k = g ≫ (Limits.limit.π H j).app k) : f = g :=
  by
  apply (cancel_mono (limit_obj_iso_limit_comp_evaluation H k).Hom).1
  ext
  simpa using w j
#align category_theory.limits.limit_obj_ext CategoryTheory.Limits.limit_obj_ext

/- warning: category_theory.limits.evaluation_preserves_colimits_of_shape -> CategoryTheory.Limits.evaluationPreservesColimitsOfShape is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_colimits_of_shape CategoryTheory.Limits.evaluationPreservesColimitsOfShapeₓ'. -/
instance evaluationPreservesColimitsOfShape [HasColimitsOfShape J C] (k : K) :
    PreservesColimitsOfShape J ((evaluation K C).obj k)
    where PreservesColimit F :=
    preservesColimitOfPreservesColimitCocone (combinedIsColimit _ _) <|
      IsColimit.ofIsoColimit (colimit.isColimit _) (evaluateCombinedCocones F _ k).symm
#align category_theory.limits.evaluation_preserves_colimits_of_shape CategoryTheory.Limits.evaluationPreservesColimitsOfShape

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation CategoryTheory.Limits.colimitObjIsoColimitCompEvaluationₓ'. -/
/-- If `F : J ⥤ K ⥤ C` is a functor into a functor category which has a colimit,
then the evaluation of that colimit at `k` is the colimit of the evaluations of `F.obj j` at `k`.
-/
def colimitObjIsoColimitCompEvaluation [HasColimitsOfShape J C] (F : J ⥤ K ⥤ C) (k : K) :
    (colimit F).obj k ≅ colimit (F ⋙ (evaluation K C).obj k) :=
  preservesColimitIso ((evaluation K C).obj k) F
#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_inv CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_ι_invₓ'. -/
@[simp, reassoc.1]
theorem colimitObjIsoColimitCompEvaluation_ι_inv [HasColimitsOfShape J C] (F : J ⥤ K ⥤ C) (j : J)
    (k : K) :
    colimit.ι (F ⋙ (evaluation K C).obj k) j ≫ (colimitObjIsoColimitCompEvaluation F k).inv =
      (colimit.ι F j).app k :=
  by
  dsimp [colimit_obj_iso_colimit_comp_evaluation]
  simp
#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_inv CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_ι_inv

/- warning: category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_app_hom -> CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_ι_app_hom is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_app_hom CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_ι_app_homₓ'. -/
@[simp, reassoc.1]
theorem colimitObjIsoColimitCompEvaluation_ι_app_hom [HasColimitsOfShape J C] (F : J ⥤ K ⥤ C)
    (j : J) (k : K) :
    (colimit.ι F j).app k ≫ (colimitObjIsoColimitCompEvaluation F k).Hom =
      colimit.ι (F ⋙ (evaluation K C).obj k) j :=
  by
  dsimp [colimit_obj_iso_colimit_comp_evaluation]
  rw [← iso.eq_comp_inv]
  simp
#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_app_hom CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_ι_app_hom

/- warning: category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map -> CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map is a dubious translation:
lean 3 declaration is
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(CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) 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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_mapₓ'. -/
@[simp, reassoc.1]
theorem colimitObjIsoColimitCompEvaluation_inv_colimit_map [HasColimitsOfShape J C] (F : J ⥤ K ⥤ C)
    {i j : K} (f : i ⟶ j) :
    (colimitObjIsoColimitCompEvaluation _ _).inv ≫ (colimit F).map f =
      colimMap (whiskerLeft _ ((evaluation _ _).map f)) ≫
        (colimitObjIsoColimitCompEvaluation _ _).inv :=
  by
  ext
  dsimp
  simp
#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map

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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom CategoryTheory.Limits.colimit_map_colimitObjIsoColimitCompEvaluation_homₓ'. -/
@[simp, reassoc.1]
theorem colimit_map_colimitObjIsoColimitCompEvaluation_hom [HasColimitsOfShape J C] (F : J ⥤ K ⥤ C)
    {i j : K} (f : i ⟶ j) :
    (colimit F).map f ≫ (colimitObjIsoColimitCompEvaluation _ _).Hom =
      (colimitObjIsoColimitCompEvaluation _ _).Hom ≫
        colimMap (whiskerLeft _ ((evaluation _ _).map f)) :=
  by
  rw [← iso.inv_comp_eq, ← category.assoc, ← iso.eq_comp_inv,
    colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map]
#align category_theory.limits.colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom CategoryTheory.Limits.colimit_map_colimitObjIsoColimitCompEvaluation_hom

/- warning: category_theory.limits.colimit_obj_ext -> CategoryTheory.Limits.colimit_obj_ext is a dubious translation:
lean 3 declaration is
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_inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k) g)) -> (Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) W) f g)
Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_ext CategoryTheory.Limits.colimit_obj_extₓ'. -/
@[ext]
theorem colimit_obj_ext {H : J ⥤ K ⥤ C} [HasColimitsOfShape J C] {k : K} {W : C}
    {f g : (colimit H).obj k ⟶ W} (w : ∀ j, (colimit.ι H j).app k ≫ f = (colimit.ι H j).app k ≫ g) :
    f = g := by
  apply (cancel_epi (colimit_obj_iso_colimit_comp_evaluation H k).inv).1
  ext
  simpa using w j
#align category_theory.limits.colimit_obj_ext CategoryTheory.Limits.colimit_obj_ext

/- warning: category_theory.limits.evaluation_preserves_limits -> CategoryTheory.Limits.evaluationPreservesLimits is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_limits CategoryTheory.Limits.evaluationPreservesLimitsₓ'. -/
instance evaluationPreservesLimits [HasLimits C] (k : K) : PreservesLimits ((evaluation K C).obj k)
    where PreservesLimitsOfShape J 𝒥 := by skip <;> infer_instance
#align category_theory.limits.evaluation_preserves_limits CategoryTheory.Limits.evaluationPreservesLimits

/- warning: category_theory.limits.preserves_limit_of_evaluation -> CategoryTheory.Limits.preservesLimitOfEvaluation is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limit_of_evaluation CategoryTheory.Limits.preservesLimitOfEvaluationₓ'. -/
/-- `F : D ⥤ K ⥤ C` preserves the limit of some `G : J ⥤ D` if it does for each `k : K`. -/
def preservesLimitOfEvaluation (F : D ⥤ K ⥤ C) (G : J ⥤ D)
    (H : ∀ k : K, PreservesLimit G (F ⋙ (evaluation K C).obj k : D ⥤ C)) : PreservesLimit G F :=
  ⟨fun c hc => by
    apply evaluation_jointly_reflects_limits
    intro X
    haveI := H X
    change is_limit ((F ⋙ (evaluation K C).obj X).mapCone c)
    exact preserves_limit.preserves hc⟩
#align category_theory.limits.preserves_limit_of_evaluation CategoryTheory.Limits.preservesLimitOfEvaluation

/- warning: category_theory.limits.preserves_limits_of_shape_of_evaluation -> CategoryTheory.Limits.preservesLimitsOfShapeOfEvaluation is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limits_of_shape_of_evaluation CategoryTheory.Limits.preservesLimitsOfShapeOfEvaluationₓ'. -/
/-- `F : D ⥤ K ⥤ C` preserves limits of shape `J` if it does for each `k : K`. -/
def preservesLimitsOfShapeOfEvaluation (F : D ⥤ K ⥤ C) (J : Type _) [Category J]
    (H : ∀ k : K, PreservesLimitsOfShape J (F ⋙ (evaluation K C).obj k)) :
    PreservesLimitsOfShape J F :=
  ⟨fun G => preservesLimitOfEvaluation F G fun k => PreservesLimitsOfShape.preservesLimit⟩
#align category_theory.limits.preserves_limits_of_shape_of_evaluation CategoryTheory.Limits.preservesLimitsOfShapeOfEvaluation

/- warning: category_theory.limits.preserves_limits_of_evaluation -> CategoryTheory.Limits.preservesLimitsOfEvaluation is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limits_of_evaluation CategoryTheory.Limits.preservesLimitsOfEvaluationₓ'. -/
/-- `F : D ⥤ K ⥤ C` preserves all limits if it does for each `k : K`. -/
def preservesLimitsOfEvaluation (F : D ⥤ K ⥤ C)
    (H : ∀ k : K, PreservesLimitsOfSize.{w', w} (F ⋙ (evaluation K C).obj k)) :
    PreservesLimitsOfSize.{w', w} F :=
  ⟨fun L hL =>
    preserves_limits_of_shape_of_evaluation F L fun k =>
      preserves_limits_of_size.preserves_limits_of_shape⟩
#align category_theory.limits.preserves_limits_of_evaluation CategoryTheory.Limits.preservesLimitsOfEvaluation

#print CategoryTheory.Limits.preservesLimitsConst /-
/-- The constant functor `C ⥤ (D ⥤ C)` preserves limits. -/
instance preservesLimitsConst : PreservesLimitsOfSize.{w', w} (const D : C ⥤ _) :=
  preservesLimitsOfEvaluation _ fun X =>
    preservesLimitsOfNatIso <| Iso.symm <| constCompEvaluationObj _ _
#align category_theory.limits.preserves_limits_const CategoryTheory.Limits.preservesLimitsConst
-/

/- warning: category_theory.limits.evaluation_preserves_colimits -> CategoryTheory.Limits.evaluationPreservesColimits is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_colimits CategoryTheory.Limits.evaluationPreservesColimitsₓ'. -/
instance evaluationPreservesColimits [HasColimits C] (k : K) :
    PreservesColimits ((evaluation K C).obj k)
    where PreservesColimitsOfShape J 𝒥 := by skip <;> infer_instance
#align category_theory.limits.evaluation_preserves_colimits CategoryTheory.Limits.evaluationPreservesColimits

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimit_of_evaluation CategoryTheory.Limits.preservesColimitOfEvaluationₓ'. -/
/-- `F : D ⥤ K ⥤ C` preserves the colimit of some `G : J ⥤ D` if it does for each `k : K`. -/
def preservesColimitOfEvaluation (F : D ⥤ K ⥤ C) (G : J ⥤ D)
    (H : ∀ k, PreservesColimit G (F ⋙ (evaluation K C).obj k)) : PreservesColimit G F :=
  ⟨fun c hc => by
    apply evaluation_jointly_reflects_colimits
    intro X
    haveI := H X
    change is_colimit ((F ⋙ (evaluation K C).obj X).mapCocone c)
    exact preserves_colimit.preserves hc⟩
#align category_theory.limits.preserves_colimit_of_evaluation CategoryTheory.Limits.preservesColimitOfEvaluation

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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimits_of_shape_of_evaluation CategoryTheory.Limits.preservesColimitsOfShapeOfEvaluationₓ'. -/
/-- `F : D ⥤ K ⥤ C` preserves all colimits of shape `J` if it does for each `k : K`. -/
def preservesColimitsOfShapeOfEvaluation (F : D ⥤ K ⥤ C) (J : Type _) [Category J]
    (H : ∀ k : K, PreservesColimitsOfShape J (F ⋙ (evaluation K C).obj k)) :
    PreservesColimitsOfShape J F :=
  ⟨fun G => preservesColimitOfEvaluation F G fun k => PreservesColimitsOfShape.preservesColimit⟩
#align category_theory.limits.preserves_colimits_of_shape_of_evaluation CategoryTheory.Limits.preservesColimitsOfShapeOfEvaluation

/- warning: category_theory.limits.preserves_colimits_of_evaluation -> CategoryTheory.Limits.preservesColimitsOfEvaluation is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimits_of_evaluation CategoryTheory.Limits.preservesColimitsOfEvaluationₓ'. -/
/-- `F : D ⥤ K ⥤ C` preserves all colimits if it does for each `k : K`. -/
def preservesColimitsOfEvaluation (F : D ⥤ K ⥤ C)
    (H : ∀ k : K, PreservesColimitsOfSize.{w', w} (F ⋙ (evaluation K C).obj k)) :
    PreservesColimitsOfSize.{w', w} F :=
  ⟨fun L hL =>
    preserves_colimits_of_shape_of_evaluation F L fun k =>
      preserves_colimits_of_size.preserves_colimits_of_shape⟩
#align category_theory.limits.preserves_colimits_of_evaluation CategoryTheory.Limits.preservesColimitsOfEvaluation

#print CategoryTheory.Limits.preservesColimitsConst /-
/-- The constant functor `C ⥤ (D ⥤ C)` preserves colimits. -/
instance preservesColimitsConst : PreservesColimitsOfSize.{w', w} (const D : C ⥤ _) :=
  preservesColimitsOfEvaluation _ fun X =>
    preservesColimitsOfNatIso <| Iso.symm <| constCompEvaluationObj _ _
#align category_theory.limits.preserves_colimits_const CategoryTheory.Limits.preservesColimitsConst
-/

open CategoryTheory.prod

#print CategoryTheory.Limits.limitIsoFlipCompLim /-
/-- The limit of a diagram `F : J ⥤ K ⥤ C` is isomorphic to the functor given by
the individual limits on objects. -/
@[simps]
def limitIsoFlipCompLim [HasLimitsOfShape J C] (F : J ⥤ K ⥤ C) : limit F ≅ F.flip ⋙ lim :=
  NatIso.ofComponents (limitObjIsoLimitCompEvaluation F) <| by tidy
#align category_theory.limits.limit_iso_flip_comp_lim CategoryTheory.Limits.limitIsoFlipCompLim
-/

#print CategoryTheory.Limits.limitFlipIsoCompLim /-
/-- A variant of `limit_iso_flip_comp_lim` where the arguemnts of `F` are flipped. -/
@[simps]
def limitFlipIsoCompLim [HasLimitsOfShape J C] (F : K ⥤ J ⥤ C) : limit F.flip ≅ F ⋙ lim :=
  (NatIso.ofComponents fun k =>
      limitObjIsoLimitCompEvaluation F.flip k ≪≫ HasLimit.isoOfNatIso (flipCompEvaluation _ _)) <|
    by tidy
#align category_theory.limits.limit_flip_iso_comp_lim CategoryTheory.Limits.limitFlipIsoCompLim
-/

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_inst_3 K _inst_4) C _inst_1) (CategoryTheory.uncurry.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1)) G))) (CategoryTheory.Limits.lim.{u1, u3, u5, u6} J _inst_3 C _inst_1 _inst_5))
Case conversion may be inaccurate. Consider using '#align category_theory.limits.limit_iso_swap_comp_lim CategoryTheory.Limits.limitIsoSwapCompLimₓ'. -/
/-- For a functor `G : J ⥤ K ⥤ C`, its limit `K ⥤ C` is given by `(G' : K ⥤ J ⥤ C) ⋙ lim`.
Note that this does not require `K` to be small.
-/
@[simps]
def limitIsoSwapCompLim [HasLimitsOfShape J C] (G : J ⥤ K ⥤ C) :
    limit G ≅ curry.obj (swap K J ⋙ uncurry.obj G) ⋙ lim :=
  limitIsoFlipCompLim G ≪≫ isoWhiskerRight (flipIsoCurrySwapUncurry _) _
#align category_theory.limits.limit_iso_swap_comp_lim CategoryTheory.Limits.limitIsoSwapCompLim

#print CategoryTheory.Limits.colimitIsoFlipCompColim /-
/-- The colimit of a diagram `F : J ⥤ K ⥤ C` is isomorphic to the functor given by
the individual colimits on objects. -/
@[simps]
def colimitIsoFlipCompColim [HasColimitsOfShape J C] (F : J ⥤ K ⥤ C) : colimit F ≅ F.flip ⋙ colim :=
  NatIso.ofComponents (colimitObjIsoColimitCompEvaluation F) <| by tidy
#align category_theory.limits.colimit_iso_flip_comp_colim CategoryTheory.Limits.colimitIsoFlipCompColim
-/

#print CategoryTheory.Limits.colimitFlipIsoCompColim /-
/-- A variant of `colimit_iso_flip_comp_colim` where the arguemnts of `F` are flipped. -/
@[simps]
def colimitFlipIsoCompColim [HasColimitsOfShape J C] (F : K ⥤ J ⥤ C) : colimit F.flip ≅ F ⋙ colim :=
  (NatIso.ofComponents fun k =>
      colimitObjIsoColimitCompEvaluation _ _ ≪≫ HasColimit.isoOfNatIso (flipCompEvaluation _ _)) <|
    by tidy
#align category_theory.limits.colimit_flip_iso_comp_colim CategoryTheory.Limits.colimitFlipIsoCompColim
-/

/- warning: category_theory.limits.colimit_iso_swap_comp_colim -> CategoryTheory.Limits.colimitIsoSwapCompColim is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_iso_swap_comp_colim CategoryTheory.Limits.colimitIsoSwapCompColimₓ'. -/
/-- For a functor `G : J ⥤ K ⥤ C`, its colimit `K ⥤ C` is given by `(G' : K ⥤ J ⥤ C) ⋙ colim`.
Note that this does not require `K` to be small.
-/
@[simps]
def colimitIsoSwapCompColim [HasColimitsOfShape J C] (G : J ⥤ K ⥤ C) :
    colimit G ≅ curry.obj (swap K J ⋙ uncurry.obj G) ⋙ colim :=
  colimitIsoFlipCompColim G ≪≫ isoWhiskerRight (flipIsoCurrySwapUncurry _) _
#align category_theory.limits.colimit_iso_swap_comp_colim CategoryTheory.Limits.colimitIsoSwapCompColim

end CategoryTheory.Limits