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 Y : Cᵒᵖ
f : X ⟶ Y
e : (J.pullback f.unop).obj ⊤ ⟶ ⊤ := homOfLE ⋯
⊢ Eq (α := P.obj X ⟶ Functor.obj (J.diagram P (unop Y)) (op ((J.pullback f.unop).obj ⊤)))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
      (P.map f ≫
        Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (Functor.map (J.diagram P (unop Y)) e.op))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
        (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
        (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
        (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I)) ⋯))

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 Y : Cᵒᵖ
f : X ⟶ Y
e : (J.pullback f.unop).obj ⊤ ⟶ ⊤ := homOfLE ⋯
e_a :
  ∀ (I : (unop (op ((J.pullback f.unop).obj ⊤))).shape.L),
    CategoryStruct.comp (Y := multiequalizer ((unop (op ((J.pullback f.unop).obj ⊤))).index P))
        (CategoryStruct.comp (Y :=
          multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
          (P.map f ≫
            Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
              (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
          (Functor.map (J.diagram P (unop Y)) e.op))
        (Multiequalizer.ι ((unop (op ((J.pullback f.unop).obj ⊤))).index P) I) =
      CategoryStruct.comp (Y := multiequalizer ((unop (op ((J.pullback f.unop).obj ⊤))).index P))
        (CategoryStruct.comp (Y :=
          multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
          (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
            (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
          (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
            (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
            (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I))
            ⋯))
        (Multiequalizer.ι ((unop (op ((J.pullback f.unop).obj ⊤))).index P) I)
⊢ Eq (α := P.obj X ⟶ Functor.obj (J.diagram P (unop Y)) (op ((J.pullback f.unop).obj ⊤)))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
      (P.map f ≫
        Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (Functor.map (J.diagram P (unop Y)) e.op))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
        (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
        (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
        (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I)) ⋯))

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 Y : Cᵒᵖ
f : X ⟶ Y
e : (J.pullback f.unop).obj ⊤ ⟶ ⊤ := homOfLE ⋯
I : (unop (op ((J.pullback f.unop).obj ⊤))).shape.L
e_a :
  P.map f ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (CategoryStruct.comp (Y := multiequalizer (((J.pullback f.unop).obj ⊤).index P))
          (Functor.map (J.diagram P (unop Y)) e.op) (Multiequalizer.ι (((J.pullback f.unop).obj ⊤).index P) I)) =
    CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
        (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (CategoryStruct.comp (Y := multiequalizer (((J.pullback f.unop).obj ⊤).index P))
        (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
          (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
          (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I)) ⋯)
        (Multiequalizer.ι (((J.pullback f.unop).obj ⊤).index P) I))
⊢ CategoryStruct.comp (Y := multiequalizer ((unop (op ((J.pullback f.unop).obj ⊤))).index P))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
        (P.map f ≫
          Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
            (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (Functor.map (J.diagram P (unop Y)) e.op))
      (Multiequalizer.ι ((unop (op ((J.pullback f.unop).obj ⊤))).index P) I) =
    CategoryStruct.comp (Y := multiequalizer ((unop (op ((J.pullback f.unop).obj ⊤))).index P))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
          (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
          (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I)) ⋯))
      (Multiequalizer.ι ((unop (op ((J.pullback f.unop).obj ⊤))).index P) I)

Tactic:
simp only [Category.assoc]

Premises used: CategoryTheory.Category.assoc
-- 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 : Cᵒᵖ
f : X ⟶ Y
e : (J.pullback f.unop).obj ⊤ ⟶ ⊤ := homOfLE ⋯
I : (unop (op ((J.pullback f.unop).obj ⊤))).shape.L
⊢ P.map f ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj Y)
            (fun (I : Cover.Arrow.{v, u} (C := C) (X := unop Y) (J := J) ⊤) ↦ P.map I.f.op) ⋯ :
          P.obj Y ⟶ multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
        (CategoryStruct.comp (Y :=
            multiequalizer (J := Cover.shape (J := J) (Cover.pullback ⊤ f.unop))
              (Cover.index (Cover.pullback ⊤ f.unop) P))
            (Multiequalizer.lift (Cover.index (Cover.pullback ⊤ f.unop) P)
              (multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
              (fun (I : Cover.Arrow (J := J) (Cover.pullback ⊤ f.unop)) ↦ Multiequalizer.ι (Cover.index ⊤ P) (I.map e))
              ⋯)
            (Multiequalizer.ι (Cover.index (Cover.pullback ⊤ f.unop) P) I) :
          multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P) ⟶
            P.obj (op I.Y)) =
    CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X)
        (fun (I : Cover.Arrow.{v, u} (C := C) (X := unop X) (J := J) ⊤) ↦ P.map I.f.op) ⋯)
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape (J := J) (Cover.pullback ⊤ f.unop)) (Cover.index (Cover.pullback ⊤ f.unop) P))
        (Multiequalizer.lift (Cover.index (Cover.pullback ⊤ f.unop) P)
          (multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
          (fun (I : Cover.Arrow (J := J) (Cover.pullback ⊤ f.unop)) ↦
            Multiequalizer.ι (Cover.index ⊤ P) { Y := I.Y, f := I.f ≫ f.unop, hf := ⋯ })
          ⋯)
        (Multiequalizer.ι (Cover.index (Cover.pullback ⊤ f.unop) 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 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
w :
  ∀ (x : Cᵒᵖ),
    Eq (α := P.obj x ⟶ (J.plusObj Q).obj x) (NatTrans.app (η ≫ J.toPlus Q) x)
      (NatTrans.app (J.toPlus P ≫ J.plusMap η) x)
⊢ Eq (α := P ⟶ J.plusObj Q) (η ≫ J.toPlus Q) (J.toPlus P ≫ J.plusMap η)

Tactic:
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 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
w :
  Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (η.app x ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
        (Cover.toMultiequalizer ⊤ Q) (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y := (J.plusObj P).obj x)
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ P))
        (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop x)) (op ⊤)))
      (colimMap (J.diagramNatTrans η (unop x))))
⊢ Eq (α := P.obj x ⟶ (J.plusObj Q).obj x) (NatTrans.app (η ≫ J.toPlus Q) x) (NatTrans.app (J.toPlus P ≫ J.plusMap η) x)

Tactic:
dsimp [toPlus, plusMap]

Premises used: CategoryTheory.GrothendieckTopology.plusMap, CategoryTheory.GrothendieckTopology.toPlus
-- 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 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
w :
  Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (η.app x ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
        (Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
        (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y := (J.plusObj P).obj x)
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (colimit.ι (J.diagram P (unop x)) (op ⊤)))
      (colimMap (J.diagramNatTrans η (unop x))))
⊢ Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (η.app x ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
        (Cover.toMultiequalizer ⊤ Q) (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y := (J.plusObj P).obj x)
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ P))
        (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop x)) (op ⊤)))
      (colimMap (J.diagramNatTrans η (unop x))))

Tactic:
delta Cover.toMultiequalizer

Premises used: CategoryTheory.GrothendieckTopology.Cover.toMultiequalizer
-- 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 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
w :
  Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (η.app x ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
        (Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
        (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
        (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤) ≫ colimit.ι (J.diagram Q (unop x)) (op ⊤)))
⊢ Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (η.app x ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
        (Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
        (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y := (J.plusObj P).obj x)
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (colimit.ι (J.diagram P (unop x)) (op ⊤)))
      (colimMap (J.diagramNatTrans η (unop x))))

Tactic:
simp only [ι_colimMap, Category.assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.ι_colimMap
-- 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 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
w :
  Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
      (η.app x ≫
        Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
      (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y := Functor.obj (C := (J.Cover (unop x))ᵒᵖ) (J.diagram Q (unop x)) (op ⊤))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯ ≫
        NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤))
      (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
⊢ Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (η.app x ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
        (Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
        (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
        (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤) ≫ colimit.ι (J.diagram Q (unop x)) (op ⊤)))

Tactic:
simp_rw [← Category.assoc]

Premises used: CategoryTheory.Category.assoc
-- 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 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
w :
  Eq (α := P.obj x ⟶ multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
    (η.app x ≫
        Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯ :
      P.obj x ⟶ multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
    (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯ ≫
        NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤) :
      P.obj x ⟶ Functor.obj (C := (J.Cover (unop x))ᵒᵖ) (J.diagram Q (unop x)) (op ⊤))
⊢ Eq (α := P.obj x ⟶ (J.plusObj Q).obj x)
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
      (η.app x ≫
        Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
      (colimit.ι (J.diagram Q (unop x)) (op ⊤)))
    (CategoryStruct.comp (Y := Functor.obj (C := (J.Cover (unop x))ᵒᵖ) (J.diagram Q (unop x)) (op ⊤))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯ ≫
        NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤))
      (colimit.ι (J.diagram Q (unop x)) (op ⊤)))

Tactic:
congr 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L
⊢ CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
      (η.app x ≫
        Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯)
      (Multiequalizer.ι (Cover.index ⊤ Q) I) =
    CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
            (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯ ≫
          NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤) :
        P.obj x ⟶ Functor.obj (C := (J.Cover (unop x))ᵒᵖ) (J.diagram Q (unop x)) (op ⊤))
      (Multiequalizer.ι (Cover.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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
x : Cᵒᵖ
⊢ Eq (α := P.obj x ⟶ multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
    (η.app x ≫
        Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ Q.map I.f.op) ⋯ :
      P.obj x ⟶ multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤) (Cover.index ⊤ Q))
    (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop x)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop x) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯ ≫
        NatTrans.app (J.diagramNatTrans η (unop x)) (op ⊤) :
      P.obj x ⟶ Functor.obj (C := (J.Cover (unop x))ᵒᵖ) (J.diagram Q (unop x)) (op ⊤))

Tactic:
exact Multiequalizer.hom_ext _ _ _ (fun I => by simp)

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
w : ∀ (X : Cᵒᵖ), NatTrans.app (J.plusMap (J.toPlus P)) X = NatTrans.app (J.toPlus (J.plusObj P)) X
⊢ J.plusMap (J.toPlus P) = J.toPlus (J.plusObj P)

Tactic:
ext X : 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 : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
⊢ NatTrans.app (J.plusMap (J.toPlus P)) X = NatTrans.app (J.toPlus (J.plusObj P)) X

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

Premises used: CategoryTheory.Limits.colimit, 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
w :
  ∀ (S : (J.Cover (unop X))ᵒᵖ),
    Eq (α := Functor.obj (J.diagram P (unop X)) S ⟶ (J.plusObj (J.plusObj P)).obj X)
      (colimit.ι (J.diagram P (unop X)) S ≫ NatTrans.app (J.plusMap (J.toPlus P)) X)
      (colimit.ι (J.diagram P (unop X)) S ≫ NatTrans.app (J.toPlus (J.plusObj P)) X)
⊢ NatTrans.app (J.plusMap (J.toPlus P)) X = NatTrans.app (J.toPlus (J.plusObj P)) X

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

Premises used: CategoryTheory.Limits.colimit.hom_ext