awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
rw :
  Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y :=
          Functor.obj
            (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
              { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
                map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
                map_id := ⋯, map_comp := ⋯ }
              (J.diagram P (unop X)))
            S)
          (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
            (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
              Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
            ⋯)
          (Functor.map (J.diagram P (unop X)) e.inv.op) ≫
        colimit.ι (J.diagram P (unop X)) (op (unop S)) :
      Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := ⋯, map_comp := ⋯ }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (colimit.ι (J.diagram P (unop X)) (op ((unop S).pullback (𝟙 (unop X))))) :
      Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)

Tactic:
rw [← colimit.w _ e.inv.op, ← Category.assoc]

Premises used: CategoryTheory.Category.assoc, Quiver.Hom.op, CategoryTheory.Limits.colimit.w, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
⊢ Eq (α := Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (𝟙 (Functor.obj (J.diagram P (unop X)) S) ≫ colimit.ι (J.diagram P (unop X)) S) (colimit.ι (J.diagram P (unop X)) S)

Tactic:
Category.id_comp (colimit.ι (diagram J P (unop X)) S)

Premises used: Opposite.unop, CategoryTheory.GrothendieckTopology.diagram, CategoryTheory.Category.id_comp, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
h :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
      (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) →
    CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := ⋯, map_comp := ⋯ }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (Functor.map (J.diagram P (unop X)) e.inv.op) =
      𝟙 (Functor.obj (J.diagram P (unop X)) S)
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y :=
          Functor.obj
            (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
              { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
                map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
                map_id := ⋯, map_comp := ⋯ }
              (J.diagram P (unop X)))
            S)
          (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
            (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
              Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
            ⋯)
          (Functor.map (J.diagram P (unop X)) e.inv.op) ≫
        colimit.ι (J.diagram P (unop X)) (op (unop S)) :
      Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)

Tactic:
convert! Category.id_comp (colimit.ι (diagram J P (unop X)) S)

Premises used: Opposite.unop, CategoryTheory.GrothendieckTopology.diagram, CategoryTheory.Category.id_comp, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
⊢ CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
            map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
            map_id := ⋯, map_comp := ⋯ }
          (J.diagram P (unop X)))
        S)
      (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        ⋯)
      (Functor.map (J.diagram P (unop X)) e.inv.op) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S)

Tactic:
Multiequalizer.hom_ext _ _ _ (fun I => ?_)

Premises used: CategoryTheory.Limits.limit, CategoryTheory.Limits.Multiequalizer.hom_ext
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
h :
  ∀ (I : (unop (op (unop S))).shape.L),
    CategoryStruct.comp (Y := multiequalizer ((unop (op (unop S))).index P))
        (CategoryStruct.comp (Y :=
          Functor.obj
            (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
              { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
                map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
                map_id := ⋯, map_comp := ⋯ }
              (J.diagram P (unop X)))
            S)
          (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
            (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
              Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
            ⋯)
          (Functor.map (J.diagram P (unop X)) e.inv.op))
        (Multiequalizer.ι ((unop (op (unop S))).index P) I) =
      𝟙 (Functor.obj (J.diagram P (unop X)) S) ≫ Multiequalizer.ι ((unop (op (unop S))).index P) I
⊢ CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
            map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
            map_id := ⋯, map_comp := ⋯ }
          (J.diagram P (unop X)))
        S)
      (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        ⋯)
      (Functor.map (J.diagram P (unop X)) e.inv.op) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S)

Tactic:
refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)

Premises used: CategoryTheory.Limits.Multiequalizer.hom_ext
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
h :
  CategoryStruct.comp (Y := multiequalizer ((unop S).index P))
      (CategoryStruct.comp (Y := multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (multiequalizer ((unop S).index P))
          (fun (I : ((unop S).pullback (𝟙 (unop X))).Arrow) ↦ Multiequalizer.ι ((unop S).index P) I.base) ⋯)
        (Multiequalizer.lift ((unop S).index P) (multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
          (fun (I : (unop S).Arrow) ↦ Multiequalizer.ι (((unop S).pullback (𝟙 (unop X))).index P) (I.map e.inv)) ⋯))
      (Multiequalizer.ι ((unop S).index P) I) =
    𝟙 (multiequalizer ((unop S).index P)) ≫ Multiequalizer.ι ((unop S).index P) I
⊢ CategoryStruct.comp (Y := multiequalizer ((unop (op (unop S))).index P))
      (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := ⋯, map_comp := ⋯ }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (Functor.map (J.diagram P (unop X)) e.inv.op))
      (Multiequalizer.ι ((unop (op (unop S))).index P) I) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S) ≫ Multiequalizer.ι ((unop (op (unop S))).index P) I

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
h :
  Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) (Cover.Arrow.map I e.inv).base) (Multiequalizer.ι ((unop S).index P) I)
⊢ CategoryStruct.comp (Y := multiequalizer ((unop S).index P))
      (CategoryStruct.comp (Y := multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (multiequalizer ((unop S).index P))
          (fun (I : ((unop S).pullback (𝟙 (unop X))).Arrow) ↦ Multiequalizer.ι ((unop S).index P) I.base) ⋯)
        (Multiequalizer.lift ((unop S).index P) (multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
          (fun (I : (unop S).Arrow) ↦ Multiequalizer.ι (((unop S).pullback (𝟙 (unop X))).index P) (I.map e.inv)) ⋯))
      (Multiequalizer.ι ((unop S).index P) I) =
    𝟙 (multiequalizer ((unop S).index P)) ≫ Multiequalizer.ι ((unop S).index P) I

Tactic:
simp only [Multiequalizer.lift_ι, Category.id_comp, Category.assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.Multiequalizer.lift_ι, CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
h :
  Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) { Y := I.Y, f := I.f ≫ 𝟙 (unop X), hf := ⋯ })
    (Multiequalizer.ι ((unop S).index P) I)
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) (Cover.Arrow.map I e.inv).base) (Multiequalizer.ι ((unop S).index P) I)

Tactic:
dsimp [Cover.Arrow.map, Cover.Arrow.base]

Premises used: CategoryTheory.GrothendieckTopology.Cover.Arrow.base, CategoryTheory.GrothendieckTopology.Cover.Arrow.map
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
h :
  ∀ (Y : C) (f : Y ⟶ unop X) (hf : (↑(unop (op (unop S))) : Sieve (unop X)).arrows f),
    Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y))
      (Multiequalizer.ι ((unop S).index P)
        { Y := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y,
          f := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X), hf := ⋯ })
      (Multiequalizer.ι ((unop S).index P) { Y := Y, f := f, hf := hf })
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) { Y := I.Y, f := I.f ≫ 𝟙 (unop X), hf := ⋯ })
    (Multiequalizer.ι ((unop S).index P) I)

Tactic:
cases I
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
Y : C
f : Y ⟶ unop X
hf : (↑(unop (op (unop S))) : Sieve (unop X)).arrows f
h : { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X) = f
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y))
    (Multiequalizer.ι ((unop S).index P)
      { Y := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y,
        f := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X), hf := ⋯ })
    (Multiequalizer.ι ((unop S).index P) { Y := Y, f := f, hf := hf })

Tactic:
congr
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
Y : C
f : Y ⟶ unop X
hf : (↑(unop (op (unop S))) : Sieve (unop X)).arrows f
⊢ { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X) = f

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
⊢ Eq (α :=
    colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) ⟶
      colimit (J := (J.Cover (unop Z))ᵒᵖ) (J.diagram P (unop Z)))
    (CategoryStruct.comp (Y := colimit ((J.pullback (f ≫ g).unop).op ⋙ J.diagram P (unop Z)))
      (colimMap (J.diagramPullback P (f ≫ g).unop)) (colimit.pre (J.diagram P (unop Z)) (J.pullback (f ≫ g).unop).op))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y)))
      (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram P (unop Y)))
        (colimMap (J.diagramPullback P f.unop)) (colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op))
      (CategoryStruct.comp (Y := colimit ((J.pullback g.unop).op ⋙ J.diagram P (unop Z)))
        (colimMap (J.diagramPullback P g.unop)) (colimit.pre (J.diagram P (unop Z)) (J.pullback g.unop).op)))

Tactic:
colimit.hom_ext (fun S => ?_)

Premises used: CategoryTheory.Limits.colimit.hom_ext
-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
S : (J.Cover (unop X))ᵒᵖ
dsimp :
  Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop Z))ᵒᵖ) (J.diagram P (unop Z)))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y := colimit ((J.pullback (g.unop ≫ f.unop)).op ⋙ J.diagram P (unop Z)))
        (colimMap (J.diagramPullback P (g.unop ≫ f.unop)))
        (colimit.pre (J.diagram P (unop Z)) (J.pullback (g.unop ≫ f.unop)).op)))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y)))
        (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram P (unop Y)))
          (colimMap (J.diagramPullback P f.unop)) (colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op))
        (CategoryStruct.comp (Y := colimit ((J.pullback g.unop).op ⋙ J.diagram P (unop Z)))
          (colimMap (J.diagramPullback P g.unop)) (colimit.pre (J.diagram P (unop Z)) (J.pullback g.unop).op))))
⊢ Eq (α := Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop Z))ᵒᵖ) (J.diagram P (unop Z)))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y := colimit ((J.pullback (f ≫ g).unop).op ⋙ J.diagram P (unop Z)))
        (colimMap (J.diagramPullback P (f ≫ g).unop))
        (colimit.pre (J.diagram P (unop Z)) (J.pullback (f ≫ g).unop).op)))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y)))
        (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram P (unop Y)))
          (colimMap (J.diagramPullback P f.unop)) (colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op))
        (CategoryStruct.comp (Y := colimit ((J.pullback g.unop).op ⋙ J.diagram P (unop Z)))
          (colimMap (J.diagramPullback P g.unop)) (colimit.pre (J.diagram P (unop Z)) (J.pullback g.unop).op))))

Tactic:
dsimp