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)
X : C
K : Type s
inst_3 : SmallCategory K
inst_4 : HasLimitsOfShape.{s, s, t, w} K D
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
W : (J.Cover X)ᵒᵖ
E : Cone (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
k : K
⊢ CategoryStruct.comp (Y := multiequalizer ((unop W).index (limit F)))
      (liftToDiagramLimitObj F E : E.pt ⟶ (J.diagram (limit F) X).obj W)
      (Multiequalizer.lift ((unop W).index (F.obj k)) (multiequalizer ((unop W).index (limit F)))
          (fun (x : (unop W).Arrow) ↦
            Multiequalizer.ι ((unop W).index (limit F)) x ≫ NatTrans.app (limit.π F k) (op x.Y))
          ⋯ :
        multiequalizer ((unop W).index (limit F)) ⟶ multiequalizer ((unop W).index (F.obj k))) =
    E.π.app k
have refine : ∀ (a : (unop W).shape.L),
  CategoryStruct.comp (Y := multiequalizer ((unop W).index (F.obj k)))
      (CategoryStruct.comp (Y := multiequalizer ((unop W).index (limit F)))
        (liftToDiagramLimitObj F E : E.pt ⟶ (J.diagram (limit F) X).obj W)
        (Multiequalizer.lift ((unop W).index (F.obj k)) (multiequalizer ((unop W).index (limit F)))
            (fun (x : (unop W).Arrow) ↦
              Multiequalizer.ι ((unop W).index (limit F)) x ≫ NatTrans.app (limit.π F k) (op x.Y))
            sorry :
          multiequalizer ((unop W).index (limit F)) ⟶ multiequalizer ((unop W).index (F.obj k))))
      (Multiequalizer.ι ((unop W).index (F.obj k)) a) =
    E.π.app k ≫ Multiequalizer.ι ((unop W).index (F.obj k)) a
Tactic: refine Multiequalizer.hom_ext _ _ _ (fun a => ?_)
-- 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)
X : C
K : Type s
inst_3 : SmallCategory K
inst_4 : HasLimitsOfShape.{s, s, t, w} K D
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
W : (J.Cover X)ᵒᵖ
E : Cone (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
m :
  E.pt ⟶
    Cone.pt (F := F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
      (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫
        NatTrans.app
          (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F)))
          j =
      E.π.app j
⊢ m = liftToDiagramLimitObj F E
have refine : ∀ (a : (unop W).shape.L) (j : K),
  (m ≫ Multiequalizer.ι ((unop W).index (limit.cone F).pt) a) ≫ NatTrans.app (limit.π F j) (op a.Y) =
    CategoryStruct.comp (Y := multiequalizer ((unop W).index (limit.cone F).pt))
        (liftToDiagramLimitObj F E : E.pt ⟶ (J.diagram (limit F) X).obj W)
        (Multiequalizer.ι ((unop W).index (limit.cone F).pt) a :
          multiequalizer ((unop W).index (limit.cone F).pt) ⟶ ((unop W).index (limit.cone F).pt).left a) ≫
      NatTrans.app (limit.π F j) (op a.Y)
Tactic: refine Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => ?_))
-- 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)
X : C
K : Type s
inst_3 : SmallCategory K
inst_4 : HasLimitsOfShape.{s, s, t, w} K D
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
W : (J.Cover X)ᵒᵖ
E : Cone (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
m :
  E.pt ⟶
    Cone.pt (F := F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
      (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫
        NatTrans.app
          (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F)))
          j =
      E.π.app j
a : (unop W).shape.L
j : K
⊢ (m ≫ Multiequalizer.ι ((unop W).index (limit.cone F).pt) a) ≫ NatTrans.app (limit.π F j) (op a.Y) =
    CategoryStruct.comp (Y := multiequalizer ((unop W).index (limit.cone F).pt))
        (liftToDiagramLimitObj F E : E.pt ⟶ (J.diagram (limit F) X).obj W)
        (Multiequalizer.ι ((unop W).index (limit.cone F).pt) a :
          multiequalizer ((unop W).index (limit.cone F).pt) ⟶ ((unop W).index (limit.cone F).pt).left a) ≫
      NatTrans.app (limit.π F j) (op a.Y)
have dsimp : (m ≫ Multiequalizer.ι ((unop W).index (limit F)) a) ≫ NatTrans.app (limit.π F j) (op a.Y) =
  CategoryStruct.comp (Y := multiequalizer ((unop W).index (limit F)))
      (Multiequalizer.lift ((unop W).index (limit F)) E.pt (liftToDiagramLimitObjAux F E) sorry)
      (Multiequalizer.ι ((unop W).index (limit F)) a) ≫
    NatTrans.app (limit.π F j) (op a.Y)
Tactic: dsimp [liftToDiagramLimitObj]
-- 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)
X : C
K : Type s
inst_3 : SmallCategory K
inst_4 : HasLimitsOfShape.{s, s, t, w} K D
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
W : (J.Cover X)ᵒᵖ
E : Cone (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
m :
  E.pt ⟶
    Cone.pt (F := F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
      (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫
        NatTrans.app
          (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F)))
          j =
      E.π.app j
a : (unop W).shape.L
j : K
⊢ (m ≫ Multiequalizer.ι ((unop W).index (limit F)) a) ≫ NatTrans.app (limit.π F j) (op a.Y) =
    CategoryStruct.comp (Y := multiequalizer ((unop W).index (limit F)))
        (Multiequalizer.lift ((unop W).index (limit F)) E.pt (liftToDiagramLimitObjAux F E) ⋯)
        (Multiequalizer.ι ((unop W).index (limit F)) a) ≫
      NatTrans.app (limit.π F j) (op a.Y)
have rw : m ≫ Multiequalizer.ι ((unop W).index (limit F)) a ≫ NatTrans.app (limit.π F j) (op a.Y) =
  m ≫
    CategoryStruct.comp (Y := (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W).obj j)
      (NatTrans.app
        (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F))) j)
      (Multiequalizer.ι ((unop W).index (F.obj j)) a)
Tactic: rw [Multiequalizer.lift_ι, Category.assoc, liftToDiagramLimitObjAux_fac, ← hm,
            Category.assoc]
-- 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)
X : C
K : Type s
inst_3 : SmallCategory K
inst_4 : HasLimitsOfShape.{s, s, t, w} K D
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
W : (J.Cover X)ᵒᵖ
E : Cone (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
m :
  E.pt ⟶
    Cone.pt (F := F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
      (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫
        NatTrans.app
          (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F)))
          j =
      E.π.app j
a : (unop W).shape.L
j : K
⊢ m ≫ Multiequalizer.ι ((unop W).index (limit F)) a ≫ NatTrans.app (limit.π F j) (op a.Y) =
    m ≫
      CategoryStruct.comp (Y := (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W).obj j)
        (NatTrans.app
          (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F)))
          j)
        (Multiequalizer.ι ((unop W).index (F.obj j)) a)
have dsimp : m ≫ Multiequalizer.ι ((unop W).index (limit F)) a ≫ NatTrans.app (limit.π F j) (op a.Y) =
  m ≫
    CategoryStruct.comp (Y := multiequalizer ((unop W).index (F.obj j)))
      (Multiequalizer.lift ((unop W).index (F.obj j)) (multiequalizer ((unop W).index (limit F)))
        (fun (x : (unop W).Arrow) ↦ Multiequalizer.ι ((unop W).index (limit F)) x ≫ NatTrans.app (limit.π F j) (op x.Y))
        sorry)
      (Multiequalizer.ι ((unop W).index (F.obj j)) a)
Tactic: dsimp
-- 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)
X : C
K : Type s
inst_3 : SmallCategory K
inst_4 : HasLimitsOfShape.{s, s, t, w} K D
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
W : (J.Cover X)ᵒᵖ
E : Cone (F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
m :
  E.pt ⟶
    Cone.pt (F := F ⋙ J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W)
      (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫
        NatTrans.app
          (Cone.π (Functor.mapCone (J.diagramFunctor D X ⋙ Functor.obj (evaluation (J.Cover X)ᵒᵖ D) W) (limit.cone F)))
          j =
      E.π.app j
a : (unop W).shape.L
j : K
⊢ m ≫ Multiequalizer.ι ((unop W).index (limit F)) a ≫ NatTrans.app (limit.π F j) (op a.Y) =
    m ≫
      CategoryStruct.comp (Y := multiequalizer ((unop W).index (F.obj j)))
        (Multiequalizer.lift ((unop W).index (F.obj j)) (multiequalizer ((unop W).index (limit F)))
          (fun (x : (unop W).Arrow) ↦
            Multiequalizer.ι ((unop W).index (limit F)) x ≫ NatTrans.app (limit.π F j) (op x.Y))
          ⋯)
        (Multiequalizer.ι ((unop W).index (F.obj j)) a)
have rw : m ≫ Multiequalizer.ι ((unop W).index (limit F)) a ≫ NatTrans.app (limit.π F j) (op a.Y) =
  m ≫
    (Multifork.ofι ((unop W).index (F.obj j)) (multiequalizer ((unop W).index (limit F)))
            (fun (x : (unop W).Arrow) ↦
              Multiequalizer.ι ((unop W).index (limit F)) x ≫ NatTrans.app (limit.π F j) (op x.Y))
            sorry).π.app
      (WalkingMulticospan.left a)
Tactic: rw [limit.lift_π]
-- 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)
X : C
inst_3 : HasLimitsOfSize.{max t u v, max t u v, t, w} D
⊢ PreservesLimits (C := Functor.{v, t, u, w} Cᵒᵖ D) (J.diagramFunctor D X)
have preservesLimitsOfShape : autoParam
  (∀ {J_1 : Type (max (max t u) v)} [inst_4 : Category.{max (max t u) v, max (max t u) v} J_1],
    PreservesLimitsOfShape (C := Functor.{v, t, u, w} Cᵒᵖ D) J_1 (J.diagramFunctor D X))
  PreservesLimitsOfSize.preservesLimitsOfShape._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 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))
x : Type (max 0 u v) := (J.Cover X)ᵒᵖ
F' : K ⥤ (J.Cover X)ᵒᵖ ⥤ D := F ⋙ J.diagramFunctor D X
e : colimit (limit (F ⋙ J.diagramFunctor D X)) ≅ limit (colimit (F ⋙ J.diagramFunctor D X).flip) :=
  colimitLimitIso (F ⋙ J.diagramFunctor D X)
t : J.diagram (limit F) X ≅ limit (F ⋙ J.diagramFunctor D X) :=
  IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
    (limit.isLimit (F ⋙ J.diagramFunctor D X))
p : (J.plusObj (limit F)).obj (op X) ≅ colimit (limit (F ⋙ J.diagramFunctor D X)) := HasColimit.isoOfNatIso t
i j : K
f : i ⟶ j
⊢ (colimit (F ⋙ J.diagramFunctor D X).flip).map f ≫
      Iso.hom
        ((fun (k : K) ↦
            (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k :
              (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ≅
                colimit ((F ⋙ J.diagramFunctor D X).flip ⋙ Functor.obj (evaluation K D) k)))
          j) =
    Iso.hom
        ((fun (k : K) ↦
            (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k :
              (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ≅
                colimit ((F ⋙ J.diagramFunctor D X).flip ⋙ Functor.obj (evaluation K D) k)))
          i) ≫
      (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).map f
have rw : Iso.inv
      ((fun (k : K) ↦
          (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k :
            (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ≅
              colimit ((F ⋙ J.diagramFunctor D X).flip ⋙ Functor.obj (evaluation K D) k)))
        i) ≫
    (colimit (F ⋙ J.diagramFunctor D X).flip).map f =
  (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).map f ≫
    Iso.inv
      ((fun (k : K) ↦
          (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k :
            (colimit (F ⋙ J.diagramFunctor D X).flip).obj k ≅
              colimit ((F ⋙ J.diagramFunctor D X).flip ⋙ Functor.obj (evaluation K D) k)))
        j)
Tactic: rw [← Iso.eq_comp_inv, Category.assoc, ← Iso.inv_comp_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))
x : Type (max 0 u v) := (J.Cover X)ᵒᵖ
F' : K ⥤ (J.Cover X)ᵒᵖ ⥤ D := F ⋙ J.diagramFunctor D X
e : colimit (limit (F ⋙ J.diagramFunctor D X)) ≅ limit (colimit (F ⋙ J.diagramFunctor D X).flip) :=
  colimitLimitIso (F ⋙ J.diagramFunctor D X)
t : J.diagram (limit F) X ≅ limit (F ⋙ J.diagramFunctor D X) :=
  IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
    (limit.isLimit (F ⋙ J.diagramFunctor D X))
p : (J.plusObj (limit F)).obj (op X) ≅ colimit (limit (F ⋙ J.diagramFunctor D X)) := HasColimit.isoOfNatIso t
i j : K
f : i ⟶ j
w : (J.Cover (unop (op X)))ᵒᵖ
⊢ CategoryStruct.comp (Y := colimit (J.diagram (F.obj i) X))
      (colimit.ι (J.diagram (F.obj i) X) w : (J.diagram (F.obj i) X).obj w ⟶ colimit (J.diagram (F.obj i) X))
      (Iso.inv (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip i) ≫
          (colimit (F ⋙ J.diagramFunctor D X).flip).map f :
        (J.plusObj (F.obj i)).obj (op X) ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj j) =
    CategoryStruct.comp (Y := colimit (J.diagram (F.obj i) X)) (colimit.ι (J.diagram (F.obj i) X) w)
      (CategoryStruct.comp (Y := (J.plusObj (F.obj j)).obj (op X))
        (colimMap (J.diagramNatTrans (F.map f) X) : colimit (J.diagram (F.obj i) X) ⟶ colimit (J.diagram (F.obj j) X))
        (Iso.inv (colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip j) :
          (J.plusObj (F.obj j)).obj (op X) ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj j))
have erw : NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip w) i ≫ (colimit (F ⋙ J.diagramFunctor D X).flip).map f =
  NatTrans.app (J.diagramNatTrans (F.map f) X) w ≫ NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip w) j
Tactic: erw [colimit.ι_map_assoc,
          colimitObjIsoColimitCompEvaluation_ι_inv (F ⋙ J.diagramFunctor D X).flip w j,
          colimitObjIsoColimitCompEvaluation_ι_inv_assoc (F ⋙ J.diagramFunctor D X).flip w i]
-- 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))
x : Type (max 0 u v) := (J.Cover X)ᵒᵖ
F' : K ⥤ (J.Cover X)ᵒᵖ ⥤ D := F ⋙ J.diagramFunctor D X
e : colimit (limit (F ⋙ J.diagramFunctor D X)) ≅ limit (colimit (F ⋙ J.diagramFunctor D X).flip) :=
  colimitLimitIso (F ⋙ J.diagramFunctor D X)
t : J.diagram (limit F) X ≅ limit (F ⋙ J.diagramFunctor D X) :=
  IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
    (limit.isLimit (F ⋙ J.diagramFunctor D X))
p : (J.plusObj (limit F)).obj (op X) ≅ colimit (limit (F ⋙ J.diagramFunctor D X)) := HasColimit.isoOfNatIso t
i j : K
f : i ⟶ j
w : (J.Cover (unop (op X)))ᵒᵖ
⊢ NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip w) i ≫ (colimit (F ⋙ J.diagramFunctor D X).flip).map f =
    NatTrans.app (J.diagramNatTrans (F.map f) X) w ≫ NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip w) j
have rw : Eq (α := multiequalizer ((unop w).index (F.obj i)) ⟶ (colimit (F ⋙ J.diagramFunctor D X).flip).obj j)
  (((F ⋙ J.diagramFunctor D X).flip.obj w).map f ≫ NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip w) j)
  (NatTrans.app (J.diagramNatTrans (F.map f) X) w ≫ NatTrans.app (colimit.ι (F ⋙ J.diagramFunctor D X).flip w) j)
Tactic: rw [← (colimit.ι (F ⋙ J.diagramFunctor D X).flip w).naturality]
-- 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
⊢ CategoryStruct.comp (Y := (J.plusObj (limit F)).obj (op X))
      (liftToPlusObjLimitObj (FD := FD) (CD := CD) F X S : S.pt ⟶ (J.plusObj (limit F)).obj (op X))
      (NatTrans.app (J.plusMap (limit.π F k)) (op X) :
        (J.plusObj (limit F)).obj (op X) ⟶ (J.plusObj (F.obj k)).obj (op X)) =
    S.π.app k
have dsimp : CategoryStruct.comp (Y := (J.plusObj (limit F)).obj (op X))
    (CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)))
      (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) S)
      (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
        (Iso.hom
            (HasLimit.isoOfNatIso
              (Iso.symm
                (NatIso.ofComponents
                  (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) sorry))) :
          limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) ⟶
            limit (colimit (F ⋙ J.diagramFunctor D X).flip))
        (CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
            (Iso.inv (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
              limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ colimit (limit (F ⋙ J.diagramFunctor D X)))
            (Iso.inv
                (HasColimit.isoOfNatIso
                  (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
                    (limit.isLimit (F ⋙ J.diagramFunctor D X)))) :
              colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ (J.plusObj (limit F)).obj (op X)) :
          limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ (J.plusObj (limit F)).obj (op X))))
    (NatTrans.app (J.plusMap (limit.π F k)) (op X)) =
  S.π.app k
Tactic: dsimp only [liftToPlusObjLimitObj]