awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type s
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape.{s, s, t, w} K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ
⊢ Eq (α := multiequalizer ((unop j).index (limit F)) ⟶ (J.plusObj (F.obj k)).obj (op X))
    (Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F)))
          (fun (x : (unop j).Arrow) ↦
            Multiequalizer.ι ((unop j).index (limit F)) x ≫ NatTrans.app (limit.π F k) (op x.Y))
          ⋯ ≫
        colimit.ι (J.diagram (F.obj k) X) j :
      multiequalizer ((unop j).index (limit F)) ⟶ colimit (J.diagram (F.obj k) X))
    (NatTrans.app (NatTrans.app (Cone.π (Functor.mapCone (J.diagramFunctor D X) (limit.cone F))) k) j ≫
        CategoryStruct.comp (Y := (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
          (NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip j) k :
            ((F ⋙ J.diagramFunctor D X).flip.obj j).obj k ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
          (Iso.hom (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) :
            (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ⟶ (J.plusObj (F.obj k)).obj (op X)) :
      multiequalizer ((unop j).index (limit F)) ⟶ (J.plusObj (F.obj k)).obj (op X))
have e_a : Eq (α := multiequalizer ((unop j).index (F.obj k)) ⟶ colimit (J.diagram (F.obj k) X))
  (colimit.ι (J.diagram (F.obj k) X) j : (J.diagram (F.obj k) X).obj j ⟶ colimit (J.diagram (F.obj k) X))
  (CategoryStruct.comp (Y := (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
      (NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip j) k :
        ((F ⋙ J.diagramFunctor D X).flip.obj j).obj k ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
      (Iso.hom (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) :
        (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ⟶ (J.plusObj (F.obj k)).obj (op X)) :
    ((F ⋙ J.diagramFunctor D X).obj k).obj j ⟶ (J.plusObj (F.obj k)).obj (op X))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type s
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape.{s, s, t, w} K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ
⊢ Eq (α := multiequalizer ((unop j).index (F.obj k)) ⟶ colimit (J.diagram (F.obj k) X))
    (colimit.ι (J.diagram (F.obj k) X) j : (J.diagram (F.obj k) X).obj j ⟶ colimit (J.diagram (F.obj k) X))
    (CategoryStruct.comp (Y := (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
        (NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip j) k :
          ((F ⋙ J.diagramFunctor D X).flip.obj j).obj k ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
        (Iso.hom (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) :
          (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ⟶ (J.plusObj (F.obj k)).obj (op X)) :
      ((F ⋙ J.diagramFunctor D X).obj k).obj j ⟶ (J.plusObj (F.obj k)).obj (op X))
have e_a : Eq (α := (J.diagram (F.obj k) X).obj j ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
  (CategoryStruct.comp (Y := colimit (J.diagram (F.obj k) X))
      (colimit.ι (J.diagram (F.obj k) X) j : (J.diagram (F.obj k) X).obj j ⟶ colimit (J.diagram (F.obj k) X))
      (Iso.inv (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) :
        colimit (J.diagram (F.obj k) X) ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k) :
    (J.diagram (F.obj k) X).obj j ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
  (NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip j) k :
    ((F ⋙ J.diagramFunctor D X).flip.obj j).obj k ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
Tactic: rw [← Iso.comp_inv_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type s
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape.{s, s, t, w} K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : C
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ
⊢ Eq (α := (J.diagram (F.obj k) X).obj j ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
    (CategoryStruct.comp (Y := colimit (J.diagram (F.obj k) X))
        (colimit.ι (J.diagram (F.obj k) X) j : (J.diagram (F.obj k) X).obj j ⟶ colimit (J.diagram (F.obj k) X))
        (Iso.inv (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) :
          colimit (J.diagram (F.obj k) X) ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k) :
      (J.diagram (F.obj k) X).obj j ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
    (NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip j) k :
      ((F ⋙ J.diagramFunctor D X).flip.obj j).obj k ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
have e_a : Eq (α := (J.diagram (F.obj k) X).obj j ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
  (NatTrans.app
      (Cocone.ι (Functor.mapCocone (Functor.obj (evaluation K D) k) (colimit.cocone (F ⋙ J.diagramFunctor D X).flip)))
      j :
    (J.diagram (F.obj k) X).obj j ⟶
      Functor.obj
        (Functor.obj (Functor.const (J.Cover X)ᵒᵖ)
          (Cocone.pt (F := J.diagram (F.obj k) X)
            (Functor.mapCocone (Functor.obj (evaluation K D) k) (colimit.cocone (F ⋙ J.diagramFunctor D X).flip) :
              Cocone ((F ⋙ J.diagramFunctor D X).flip ⋙ Functor.obj (evaluation K D) k))))
        j)
  (NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip j) k :
    ((F ⋙ J.diagramFunctor D X).flip.obj j).obj k ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj k)
Tactic: erw [colimit.ι_desc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
⊢ PreservesLimitsOfShape (C := Functor.{v, t, u, w} Cᵒᵖ D) K (J.plusFunctor D)
have preservesLimit : autoParam (∀ {K_1 : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D}, PreservesLimit K_1 (J.plusFunctor D))
  PreservesLimitsOfShape.preservesLimit._autoParam
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
⊢ PreservesLimit F (J.plusFunctor D)
have preservesLimit : ∀ (k : Cᵒᵖ), PreservesLimit F (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) k)
Tactic: apply preservesLimit_of_evaluation
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
⊢ PreservesLimit F (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
have preservesLimit : IsLimit (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
  (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
Tactic: apply preservesLimit_of_preserves_limit_cone (limit.isLimit F)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
⊢ IsLimit (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
    (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
have preservesLimit : ∀ (s : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)) (j : K),
  CategoryStruct.comp (Y :=
      Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
        (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F)))
      ((fun (S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)) ↦
            (liftToPlusObjLimitObj (FD := FD) (CD := CD) F (unop X) S : S.pt ⟶ (J.plusObj (limit F)).obj (op (unop X))))
          s :
        s.pt ⟶ (J.plusObj (limit F)).obj (op (unop X)))
      (NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))) j :
        Functor.obj
            (Functor.obj (Functor.const K)
              (Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
                (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))))
            j ⟶
          (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X).obj j) =
    s.π.app j
Tactic: refine ⟨fun S => liftToPlusObjLimitObj F X.unop S, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
preservesLimit :
  ∀ (s : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)) (j : K),
    CategoryStruct.comp (Y :=
        Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
          (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F)))
        ((fun (S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)) ↦
              (liftToPlusObjLimitObj (FD := FD) (CD := CD) F (unop X) S :
                S.pt ⟶ (J.plusObj (limit F)).obj (op (unop X))))
            s :
          s.pt ⟶ (J.plusObj (limit F)).obj (op (unop X)))
        (NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))) j :
          Functor.obj
              (Functor.obj (Functor.const K)
                (Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
                  (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))))
              j ⟶
            (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X).obj j) =
      s.π.app j
⊢ IsLimit (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
    (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
have preservesLimit₁ : ∀ (s : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X))
  (m :
    s.pt ⟶
      Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
        (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))),
  (∀ (j : K),
      m ≫
          NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F)))
            j =
        s.π.app j) →
    m =
      (fun (S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)) ↦
          (liftToPlusObjLimitObj (FD := FD) (CD := CD) F (unop X) S : S.pt ⟶ (J.plusObj (limit F)).obj (op (unop X))))
        s
Tactic: refine ⟨fun S => liftToPlusObjLimitObj F X.unop S, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
m :
  S.pt ⟶
    Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
      (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫ NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))) j =
      S.π.app j
⊢ m =
    (fun (S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)) ↦
        (liftToPlusObjLimitObj (FD := FD) (CD := CD) F (unop X) S : S.pt ⟶ (J.plusObj (limit F)).obj (op (unop X))))
      S
have preservesLimit : m =
  CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X))
    (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) S)
    (CategoryStruct.comp (Y :=
      limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
      (Iso.hom
        (HasLimit.isoOfNatIso
          (Iso.symm
            (NatIso.ofComponents
              (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ J.diagramFunctor D (unop X))) k)
              sorry))))
      (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
        (Iso.inv (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))) :
          limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) ⟶
            colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
        (Iso.inv
            (HasColimit.isoOfNatIso
              (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F))
                (limit.isLimit (F ⋙ J.diagramFunctor D (unop X))))) :
          colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶ (J.plusObj (limit F)).obj X)))
Tactic: dsimp [liftToPlusObjLimitObj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.LeftExact (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
m :
  S.pt ⟶
    Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
      (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫ NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))) j =
      S.π.app j
k_1 : K
k : (J.Cover (unop X))ᵒᵖ
| (CategoryStruct.comp (Y := Functor.obj (limit (F ⋙ J.diagramFunctor D (unop X))) k)
      (NatTrans.app
        (Iso.hom
          (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F))
            (limit.isLimit (F ⋙ J.diagramFunctor D (unop X)))))
        k)
      (NatTrans.app (limit.π (F ⋙ J.diagramFunctor D (unop X)) k_1) k ≫
        colimit.ι (Functor.flip (F ⋙ J.diagramFunctor D (unop X)) ⋙ Functor.obj (evaluation K D) k_1) k) :
    multiequalizer ((unop k).index (limit F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
have dsimp : (CategoryStruct.comp (Y := Functor.obj (limit (F ⋙ J.diagramFunctor D (unop X))) k)
    (NatTrans.app
      (limit.lift (F ⋙ J.diagramFunctor D (unop X)) (Functor.mapCone (J.diagramFunctor D (unop X)) (limit.cone F))) k)
    (NatTrans.app (limit.π (F ⋙ J.diagramFunctor D (unop X)) k_1) k ≫
      colimit.ι (Functor.flip (F ⋙ J.diagramFunctor D (unop X)) ⋙ Functor.obj (evaluation K D) k_1) k) :
  multiequalizer ((unop k).index (limit F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
Tactic: dsimp [IsLimit.conePointUniqueUpToIso]