awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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)
⊢ (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)

Tactic:
rw [Multiequalizer.lift_ι, Category.assoc, liftToDiagramLimitObjAux_fac, ← hm,
            Category.assoc]

Premises used: CategoryTheory.GrothendieckTopology.liftToDiagramLimitObjAux_fac, CategoryTheory.Category.assoc, CategoryTheory.Limits.Multiequalizer.lift_ι
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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))
          ⋯)
        (Multiequalizer.ι ((unop W).index (F.obj j)) a)
⊢ 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:
dsimp
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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))
              ⋯).π.app
        (WalkingMulticospan.left a)
⊢ 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)

Tactic:
rw [limit.lift_π]

Premises used: CategoryTheory.Limits.limit.lift_π
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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 ≫
      (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))
              ⋯).π.app
        (WalkingMulticospan.left a)

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
⊢ ∀ {K_1 : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D}, PreservesLimit K_1 (J.diagramFunctor D X)

Tactic:
apply preservesLimit_diagramFunctor.{s, t, w, v, u}
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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
⊢ PreservesLimits (C := Functor.{v, t, u, w} Cᵒᵖ D) (J.diagramFunctor D X)

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
J_1 : Type (max (max t u) v)
preservesLimitsOfShape :
  ∀ [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)
⊢ ∀ [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)

Tactic:
_
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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)

Tactic:
apply preservesLimitsOfShape_diagramFunctor.{max t u v}
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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)
⊢ (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

Tactic:
rw [← Iso.eq_comp_inv, Category.assoc, ← Iso.inv_comp_eq]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.eq_comp_inv, CategoryTheory.Iso.inv_comp_eq
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
⊢ 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:
colimit.hom_ext (fun w => ?_)

Premises used: CategoryTheory.Limits.colimit, CategoryTheory.Limits.colimit.hom_ext
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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)
⊢ 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:
rw [← (colimit.ι (F ⋙ J.diagramFunctor D X).flip w).naturality]

Premises used: CategoryTheory.Functor.flip, CategoryTheory.GrothendieckTopology.diagramFunctor, CategoryTheory.Functor.comp, CategoryTheory.NatTrans.naturality, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
⊢ 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:
rfl
-- Module: Mathlib.CategoryTheory.Sites.LeftExact
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
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) ⋯))) :
            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
⊢ 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

Tactic:
dsimp only [liftToPlusObjLimitObj]

Premises used: CategoryTheory.GrothendieckTopology.liftToPlusObjLimitObj