l3lab/ntp-mathlib huggingface dataset:
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
⊢ m = (fun E => liftToDiagramLimitObj F E) E

Next tactic:
refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ (m ≫ Multiequalizer.ι (Cover.index W.unop (limit.cone F).pt) a) ≫ (limit.π F j).app (op a.Y) =
    ((fun E => liftToDiagramLimitObj F E) E ≫ Multiequalizer.ι (Cover.index W.unop (limit.cone F).pt) a) ≫
      (limit.π F j).app (op a.Y)

Next tactic:
delta liftToDiagramLimitObj
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit.cone F).pt) a ≫ (limit.π F j).app (op a.Y) =
    IsLimit.lift (isLimitOfPreserves ((evaluation Cᵒᵖ D).obj (op a.Y)) (limit.isLimit F))
        (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E) ≫
      (limit.π F j).app (op a.Y)

Next tactic:
change _ = (_ ≫ _) ≫ _
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          change _ = (_ ≫ _) ≫ _
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit.cone F).pt) a ≫ (limit.π F j).app (op a.Y) =
    ((IsLimit.liftConeMorphism
            (IsLimit.ofIsoLimit (limit.isLimit (F ⋙ (evaluation Cᵒᵖ D).obj (op a.Y)))
              (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k)) (op a.Y)).symm)
            (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E)).hom ≫
        ((Cones.functoriality F ((evaluation Cᵒᵖ D).obj (op a.Y))).mapIso
              (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))
                (limit.isLimit F))).hom.hom) ≫
      (limit.π F j).app (op a.Y)

Next tactic:
dsimp [evaluateCombinedCones]
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          change _ = (_ ≫ _) ≫ _
          dsimp [evaluateCombinedCones]
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit F)) a ≫ (limit.π F j).app (op a.Y) =
    ((limit.lift (F ⋙ (evaluation Cᵒᵖ D).obj (op a.Y)) (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E) ≫
          𝟙 (getLimitCone ((Functor.flip F).obj (op a.Y))).cone.pt) ≫
        (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k))).app (op a.Y)) ≫
      (limit.π F j).app (op a.Y)

Next tactic:
erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          change _ = (_ ≫ _) ≫ _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit F)) a ≫ (limit.π F j).app (op a.Y) =
    (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E).π.app j

Next tactic:
dsimp
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          change _ = (_ ≫ _) ≫ _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          dsimp
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit F)) a ≫ (limit.π F j).app (op a.Y) =
    E.π.app j ≫ Multiequalizer.ι (Cover.index W.unop (F.obj j)) a

Next tactic:
rw [← hm]
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          change _ = (_ ≫ _) ≫ _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          dsimp
          rw [← hm]
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit F)) a ≫ (limit.π F j).app (op a.Y) =
    (m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j) ≫
      Multiequalizer.ι (Cover.index W.unop (F.obj j)) a

Next tactic:
dsimp [diagramNatTrans]
instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          erw [Multiequalizer.lift_ι, Category.assoc]
          change _ = (_ ≫ _) ≫ _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          dsimp
          rw [← hm]
          dsimp [diagramNatTrans]
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ m ≫ Multiequalizer.ι (Cover.index W.unop (limit F)) a ≫ (limit.π F j).app (op a.Y) =
    (m ≫
        Multiequalizer.lift (Cover.index W.unop (F.obj j)) (multiequalizer (Cover.index W.unop (limit F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (limit F)) i ≫ (limit.π F j).app (op i.Y))
          (_ :
            ∀ (i : (Cover.index W.unop (F.obj j)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (limit F)) i ≫ (limit.π F j).app (op i.Y))
                    (MulticospanIndex.fstTo (Cover.index W.unop (F.obj j)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (F.obj j)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (limit F)) i ≫ (limit.π F j).app (op i.Y))
                    (MulticospanIndex.sndTo (Cover.index W.unop (F.obj j)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (F.obj j)) i)) ≫
      Multiequalizer.ι (Cover.index W.unop (F.obj j)) a

Next tactic:
simp
instance preservesLimitsOfShape_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] :
    PreservesLimitsOfShape K (J.diagramFunctor D X) :=
  ⟨by 

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
⊢ {K_1 : K ⥤ Cᵒᵖ ⥤ D} → PreservesLimit K_1 (diagramFunctor J D X)

Next tactic:
apply preservesLimit_diagramFunctor.{w, v, u}
instance preservesLimits_diagramFunctor (X : C) [HasLimits D] :
    PreservesLimits (J.diagramFunctor D X) := by
  

Proof state:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
inst✝ : HasLimits D
⊢ PreservesLimits (diagramFunctor J D X)

Next tactic:
constructor
instance preservesLimits_diagramFunctor (X : C) [HasLimits D] :
    PreservesLimits (J.diagramFunctor D X) := by
  constructor
  

Proof state:
case preservesLimitsOfShape
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
inst✝ : HasLimits D
⊢ autoParam
    ({J_1 : Type (max u v)} →
      [inst : Category.{max u v, max u v} J_1] → PreservesLimitsOfShape J_1 (diagramFunctor J D X))
    _auto✝

Next tactic:
intro _ _
instance preservesLimits_diagramFunctor (X : C) [HasLimits D] :
    PreservesLimits (J.diagramFunctor D X) := by
  constructor
  intro _ _
  

Proof state:
case preservesLimitsOfShape
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
inst✝¹ : HasLimits D
J✝ : Type (max u v)
inst✝ : Category.{max u v, max u v} J✝
⊢ PreservesLimitsOfShape J✝ (diagramFunctor J D X)

Next tactic:
apply preservesLimitsOfShape_diagramFunctor.{w, v, u}
/-- An auxiliary definition to be used in the proof that `J.plusFunctor D` commutes
with finite limits. -/
def liftToPlusObjLimitObj {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) :
    S.pt ⟶ (J.plusObj (limit F)).obj (op X) :=
  let e := colimitLimitIso (F ⋙ J.diagramFunctor D X)
  let t : J.diagram (limit F) X ≅ limit (F ⋙ J.diagramFunctor D X) :=
    (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F)).conePointUniqueUpToIso
      (limit.isLimit _)
  let p : (J.plusObj (limit F)).obj (op X) ≅ colimit (limit (F ⋙ J.diagramFunctor D X)) :=
    HasColimit.isoOfNatIso t
  let s :
    colimit (F ⋙ J.diagramFunctor D X).flip ≅ F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X) :=
    NatIso.ofComponents (fun k => colimitObjIsoColimitCompEvaluation _ k)
      (by
        

Proof state:
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
e : colimit (limit (F ⋙ diagramFunctor J D X)) ≅ limit (colimit (Functor.flip (F ⋙ diagramFunctor J D X))) :=
  colimitLimitIso (F ⋙ diagramFunctor J D X)
t : diagram J (limit F) X ≅ limit (F ⋙ diagramFunctor J D X) :=
  IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))
    (limit.isLimit (F ⋙ diagramFunctor J D X))
p : (plusObj J (limit F)).obj (op X) ≅ colimit (limit (F ⋙ diagramFunctor J D X)) := HasColimit.isoOfNatIso t
⊢ ∀ {X_1 Y : K} (f : X_1 ⟶ Y),
    (colimit (Functor.flip (F ⋙ diagramFunctor J D X))).map f ≫
        ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k) Y).hom =
      ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k) X_1).hom ≫
        (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)).map f

Next tactic:
intro i j f