awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.Over
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
f : Over.{v, u} X
⊢ ∀ {X_1 : Over.{v, u} f.left} {S : Sieve X_1},
    Membership.mem (γ := Set (Sieve (f.iteratedSliceEquiv.inverse.obj X_1)))
        (((J.over X).over f : (a : Over f) → Set (Sieve a)) (f.iteratedSliceEquiv.inverse.obj X_1))
        (Sieve.functorPushforward f.iteratedSliceEquiv.inverse S) ↔
      S ∈ (J.over f.left : (a : Over f.left) → Set (Sieve a)) X_1

Tactic:
simp [GrothendieckTopology.mem_over_iff, Sieve.overEquiv,
      ← Over.iteratedSliceBackward_forget_forget f, Sieve.functorPushforward_comp]

Premises used: CategoryTheory.Over.iteratedSliceBackward_forget_forget, CategoryTheory.Sieve.overEquiv, CategoryTheory.GrothendieckTopology.mem_over_iff, CategoryTheory.Sieve.functorPushforward_comp
-- Module: Mathlib.CategoryTheory.Sites.Over
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
f : Over.{v, u} X
⊢ f.iteratedSliceForward.IsContinuous ((J.over X).over f) (J.over f.left)

Tactic:
inferInstanceAs (f.iteratedSliceEquiv.functor.IsContinuous _ _)

Premises used: _private.0.CategoryTheory.instIsContinuousOverLeftIteratedSliceForwardOver._proof_1
-- Module: Mathlib.CategoryTheory.Sites.Over
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
f : Over.{v, u} X
⊢ f.iteratedSliceForward.IsCocontinuous ((J.over X).over f) (J.over f.left)

Tactic:
inferInstanceAs (f.iteratedSliceEquiv.functor.IsCocontinuous _ _)

Premises used: _private.0.CategoryTheory.instIsCocontinuousOverLeftIteratedSliceForwardOver._proof_1
-- Module: Mathlib.CategoryTheory.Sites.Over
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
f : Over.{v, u} X
⊢ f.iteratedSliceBackward.IsContinuous (J.over f.left) ((J.over X).over f)

Tactic:
inferInstanceAs (f.iteratedSliceEquiv.inverse.IsContinuous _ _)

Premises used: _private.0.CategoryTheory.instIsContinuousOverLeftIteratedSliceBackwardOver._proof_1
-- Module: Mathlib.CategoryTheory.Sites.Over
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
X : C
f : Over.{v, u} X
⊢ f.iteratedSliceBackward.IsCocontinuous (J.over f.left) ((J.over X).over f)

Tactic:
inferInstanceAs (f.iteratedSliceEquiv.inverse.IsCocontinuous _ _)

Premises used: _private.0.CategoryTheory.instIsCocontinuousOverLeftIteratedSliceBackwardOver._proof_1
-- 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
X Y : C
f_1 : X ⟶ Y
S T : (J.Cover Y)ᵒᵖ
f : S ⟶ T
I : (unop ((J.pullback f_1).op.obj T)).shape.L
simp :
  Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) ((Cover.Arrow.base I).map f.unop) :
      multiequalizer ((unop S).index P) ⟶ ((unop S).index P).left ((Cover.Arrow.base I).map f.unop))
    (Multiequalizer.ι ((unop S).index P) (Cover.Arrow.map I ((J.pullback f_1).map f.unop)).base)
⊢ CategoryStruct.comp (Y := multiequalizer ((unop ((J.pullback f_1).op.obj T)).index P))
      ((J.diagram P Y).map f ≫
        Multiequalizer.lift ((unop ((J.pullback f_1).op.obj T)).index P) ((J.diagram P Y).obj T)
          (fun (I : (unop ((J.pullback f_1).op.obj T)).shape.L) ↦
            Multiequalizer.ι ((unop T).index P) (Cover.Arrow.base I))
          ⋯)
      (Multiequalizer.ι ((unop ((J.pullback f_1).op.obj T)).index P) I) =
    (Multiequalizer.lift ((unop ((J.pullback f_1).op.obj S)).index P) ((J.diagram P Y).obj S)
          (fun (I : (unop ((J.pullback f_1).op.obj S)).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯ ≫
        ((J.pullback f_1).op ⋙ J.diagram P X).map f) ≫
      Multiequalizer.ι ((unop ((J.pullback f_1).op.obj T)).index P) I

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
X Y : C
f_1 : X ⟶ Y
S T : (J.Cover Y)ᵒᵖ
f : S ⟶ T
I : (unop ((J.pullback f_1).op.obj T)).shape.L
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) ((Cover.Arrow.base I).map f.unop) :
      multiequalizer ((unop S).index P) ⟶ ((unop S).index P).left ((Cover.Arrow.base I).map f.unop))
    (Multiequalizer.ι ((unop S).index P) (Cover.Arrow.map I ((J.pullback f_1).map f.unop)).base)

Tactic:
rfl
-- 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_1 P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
X : C
W : (J.Cover X)ᵒᵖ
i : (unop W).shape.R
dsimp :
  Eq (α := (J.diagram P X).obj W ⟶ ((unop W).index Q).right i)
    ((Multiequalizer.ι ((unop W).index P) ((unop W).shape.fst i) ≫ η.app (op ((unop W).shape.fst i).Y)) ≫
      ((unop W).index Q).fst i)
    ((Multiequalizer.ι ((unop W).index P) ((unop W).shape.snd i) ≫ η.app (op ((unop W).shape.snd i).Y)) ≫
      ((unop W).index Q).snd i)
⊢ Eq (α := (J.diagram P X).obj W ⟶ ((unop W).index Q).right i)
    ((fun (x : (unop W).shape.L) ↦
          (Multiequalizer.ι ((unop W).index P) x ≫ η.app (op x.Y) : (J.diagram P X).obj W ⟶ ((unop W).index Q).left x))
        ((unop W).shape.fst i) ≫
      ((unop W).index Q).fst i)
    ((fun (x : (unop W).shape.L) ↦
          (Multiequalizer.ι ((unop W).index P) x ≫ η.app (op x.Y) : (J.diagram P X).obj W ⟶ ((unop W).index Q).left x))
        ((unop W).shape.snd i) ≫
      ((unop W).index Q).snd i)

Tactic:
dsimp only
-- 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_1 P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
X : C
W : (J.Cover X)ᵒᵖ
i : (unop W).shape.R
erw :
  Eq (α := (J.diagram P X).obj W ⟶ ((unop W).index Q).right i)
    (Multiequalizer.ι ((unop W).index P) ((unop W).shape.snd i) ≫ ((unop W).index P).snd i ≫ η.app (op i.r.Z))
    (Multiequalizer.ι ((unop W).index P) ((unop W).shape.snd i) ≫ P.map i.r.g₂.op ≫ η.app (op i.r.Z))
⊢ Eq (α := (J.diagram P X).obj W ⟶ ((unop W).index Q).right i)
    ((Multiequalizer.ι ((unop W).index P) ((unop W).shape.fst i) ≫ η.app (op ((unop W).shape.fst i).Y)) ≫
      ((unop W).index Q).fst i)
    ((Multiequalizer.ι ((unop W).index P) ((unop W).shape.snd i) ≫ η.app (op ((unop W).shape.snd i).Y)) ≫
      ((unop W).index Q).snd i)

Tactic:
erw [Category.assoc, Category.assoc, ← η.naturality, ← η.naturality,
        Multiequalizer.condition_assoc]

Premises used: CategoryTheory.Limits.Multiequalizer.condition_assoc, CategoryTheory.Category.assoc, CategoryTheory.NatTrans.naturality
-- 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_1 P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
X : C
W : (J.Cover X)ᵒᵖ
i : (unop W).shape.R
⊢ Eq (α := (J.diagram P X).obj W ⟶ ((unop W).index Q).right i)
    (Multiequalizer.ι ((unop W).index P) ((unop W).shape.snd i) ≫ ((unop W).index P).snd i ≫ η.app (op i.r.Z))
    (Multiequalizer.ι ((unop W).index P) ((unop W).shape.snd i) ≫ P.map i.r.g₂.op ≫ η.app (op i.r.Z))

Tactic:
rfl
-- 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)
X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
w : ∀ (x : (J.Cover X)ᵒᵖ), NatTrans.app (J.diagramNatTrans (𝟙 P) X) x = NatTrans.app (𝟙 (J.diagram P X)) x
⊢ J.diagramNatTrans (𝟙 P) X = 𝟙 (J.diagram P X)

Tactic:
ext : 2
-- 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)
X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
x : (J.Cover X)ᵒᵖ
⊢ NatTrans.app (J.diagramNatTrans (𝟙 P) X) x = NatTrans.app (𝟙 (J.diagram P X)) x

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)
X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
x : (J.Cover X)ᵒᵖ
w :
  ∀ (i : (unop x).shape.L),
    NatTrans.app (J.diagramNatTrans (𝟙 P) X) x ≫ Multiequalizer.ι ((unop x).index P) i =
      NatTrans.app (𝟙 (J.diagram P X)) x ≫ Multiequalizer.ι ((unop x).index P) i
⊢ NatTrans.app (J.diagramNatTrans (𝟙 P) X) x = NatTrans.app (𝟙 (J.diagram P X)) x

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)
X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
x : (J.Cover X)ᵒᵖ
i : (unop x).shape.L
⊢ NatTrans.app (J.diagramNatTrans (𝟙 P) X) x ≫ Multiequalizer.ι ((unop x).index P) i =
    NatTrans.app (𝟙 (J.diagram P X)) x ≫ Multiequalizer.ι ((unop x).index P) i

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)
inst_3 : Preadditive.{w', w} D
X : C
P Q : Functor.{v, w', u, w} Cᵒᵖ D
w :
  ∀ (x : (J.Cover X)ᵒᵖ),
    Eq.{w' + 1} (α := (J.diagram P X).obj x ⟶ (J.diagram Q X).obj x) (NatTrans.app (J.diagramNatTrans 0 X) x)
      (NatTrans.app 0 x)
⊢ Eq (α := J.diagram P X ⟶ J.diagram Q X) (J.diagramNatTrans 0 X) 0

Tactic:
ext : 2
-- 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)
inst_3 : Preadditive.{w', w} D
X : C
P Q : Functor.{v, w', u, w} Cᵒᵖ D
x : (J.Cover X)ᵒᵖ
⊢ Eq.{w' + 1} (α := (J.diagram P X).obj x ⟶ (J.diagram Q X).obj x) (NatTrans.app (J.diagramNatTrans 0 X) x)
    (NatTrans.app 0 x)

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)
inst_3 : Preadditive.{w', w} D
X : C
P Q : Functor.{v, w', u, w} Cᵒᵖ D
x : (J.Cover X)ᵒᵖ
w :
  ∀ (i : (unop x).shape.L),
    Eq (α := (J.diagram P X).obj x ⟶ ((unop x).index Q).left i)
      (NatTrans.app (J.diagramNatTrans 0 X) x ≫ Multiequalizer.ι ((unop x).index Q) i)
      (CategoryStruct.comp (Y := multiequalizer ((unop x).index Q))
        (NatTrans.app 0 x : (J.diagram P X).obj x ⟶ (J.diagram Q X).obj x) (Multiequalizer.ι ((unop x).index Q) i))
⊢ Eq.{w' + 1} (α := (J.diagram P X).obj x ⟶ (J.diagram Q X).obj x) (NatTrans.app (J.diagramNatTrans 0 X) x)
    (NatTrans.app 0 x)

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)
inst_3 : Preadditive.{w', w} D
X : C
P Q : Functor.{v, w', u, w} Cᵒᵖ D
x : (J.Cover X)ᵒᵖ
i : (unop x).shape.L
⊢ Eq (α := (J.diagram P X).obj x ⟶ ((unop x).index Q).left i)
    (NatTrans.app (J.diagramNatTrans 0 X) x ≫ Multiequalizer.ι ((unop x).index Q) i)
    (CategoryStruct.comp (Y := multiequalizer ((unop x).index Q))
      (NatTrans.app 0 x : (J.diagram P X).obj x ⟶ (J.diagram Q X).obj x) (Multiequalizer.ι ((unop x).index Q) i))

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 Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
w :
  ∀ (x : (J.Cover X)ᵒᵖ),
    Eq (α := (J.diagram P X).obj x ⟶ (J.diagram R X).obj x) (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x)
      (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x)
⊢ Eq (α := J.diagram P X ⟶ J.diagram R X) (J.diagramNatTrans (η ≫ γ) X) (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X)

Tactic:
ext : 2
-- 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 Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x : (J.Cover X)ᵒᵖ
⊢ Eq (α := (J.diagram P X).obj x ⟶ (J.diagram R X).obj x) (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x)
    (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x)

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 Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x : (J.Cover X)ᵒᵖ
w :
  ∀ (i : (unop x).shape.L),
    Eq (α := (J.diagram P X).obj x ⟶ ((unop x).index R).left i)
      (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x ≫ Multiequalizer.ι ((unop x).index R) i)
      (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x ≫ Multiequalizer.ι ((unop x).index R) i)
⊢ Eq (α := (J.diagram P X).obj x ⟶ (J.diagram R X).obj x) (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x)
    (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x)

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 Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x : (J.Cover X)ᵒᵖ
i : (unop x).shape.L
⊢ Eq (α := (J.diagram P X).obj x ⟶ ((unop x).index R).left i)
    (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x ≫ Multiequalizer.ι ((unop x).index R) i)
    (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x ≫ Multiequalizer.ι ((unop x).index R) i)

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 : Cᵒᵖ
⊢ CategoryStruct.comp (Y :=
      colimit (J := (J.Cover (unop X))ᵒᵖ)
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
          (J.diagram P (unop X))))
      (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
      (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))) =
    𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))

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 : Cᵒᵖ
refine :
  ∀ (S : (J.Cover (unop X))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 X))ᵒᵖ)
            (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
              (J.diagram P (unop X))))
          (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
          (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X)))))) =
      colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
⊢ CategoryStruct.comp (Y :=
      colimit (J := (J.Cover (unop X))ᵒᵖ)
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
          (J.diagram P (unop X))))
      (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
      (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))) =
    𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))

Tactic:
refine 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 : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
dsimp :
  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 (𝟙 (unop X))).op ⋙ J.diagram P (unop X)))
        (colimMap (J.diagramPullback P (𝟙 (unop X))))
        (colimit.pre (J.diagram P (unop X)) (J.pullback (𝟙 (unop X))).op)) =
    colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
⊢ 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 X))ᵒᵖ)
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
            (J.diagram P (unop X))))
        (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
        (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X)))))) =
    colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))

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
simp :
  Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y := ((J.pullback (𝟙 (unop X))).op ⋙ J.diagram P (unop X)).obj S)
        (Multiequalizer.lift ((unop ((J.pullback (𝟙 (unop X))).op.obj S)).index P)
          (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : (unop ((J.pullback (𝟙 (unop X))).op.obj S)).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (colimit.ι (J.diagram P (unop X)) ((J.pullback (𝟙 (unop X))).op.obj 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)
⊢ 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 (𝟙 (unop X))).op ⋙ J.diagram P (unop X)))
        (colimMap (J.diagramPullback P (𝟙 (unop X))))
        (colimit.pre (J.diagram P (unop X)) (J.pullback (𝟙 (unop X))).op)) =
    colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))

Tactic:
simp only [diagramPullback_app, colimit.ι_pre, ι_colimMap_assoc, Category.comp_id]

Premises used: CategoryTheory.Limits.ι_colimMap_assoc, CategoryTheory.Limits.colimit.ι_pre, CategoryTheory.Category.comp_id, CategoryTheory.GrothendieckTopology.diagramPullback_app
-- 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
dsimp :
  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)
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y := ((J.pullback (𝟙 (unop X))).op ⋙ J.diagram P (unop X)).obj S)
        (Multiequalizer.lift ((unop ((J.pullback (𝟙 (unop X))).op.obj S)).index P)
          (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : (unop ((J.pullback (𝟙 (unop X))).op.obj S)).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (colimit.ι (J.diagram P (unop X)) ((J.pullback (𝟙 (unop X))).op.obj 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:
dsimp only [Functor.op, pullback_obj]

Premises used: CategoryTheory.Functor.op, CategoryTheory.GrothendieckTopology.pullback_obj