case w.h
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
x✝ : (Cover J X)ᵒᵖ
⊢ (diagramNatTrans J 0 X).app x✝ = 0.app x✝
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
⊢ diagramNatTrans J 0 X = 0
ext : 2
case w.h
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
x✝ : (Cover J X)ᵒᵖ
i : (Cover.index x✝.unop Q).L
⊢ (diagramNatTrans J 0 X).app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop Q) i =
    0.app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop Q) i
case w.h
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
x✝ : (Cover J X)ᵒᵖ
⊢ (diagramNatTrans J 0 X).app x✝ = 0.app x✝
refine' Multiequalizer.hom_ext _ _ _ (fun i => _)
case w.h
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
x✝ : (Cover J X)ᵒᵖ
i : (Cover.index x✝.unop Q).L
⊢ Multiequalizer.lift (Cover.index x✝.unop Q) (multiequalizer (Cover.index x✝.unop P))
        (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ 0)
        (_ :
          ∀ (i : (Cover.index x✝.unop Q).R),
            (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ 0.app i.Y.op)
                  (MulticospanIndex.fstTo (Cover.index x✝.unop Q) i) ≫
                MulticospanIndex.fst (Cover.index x✝.unop Q) i =
              (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ 0.app i.Y.op)
                  (MulticospanIndex.sndTo (Cover.index x✝.unop Q) i) ≫
                MulticospanIndex.snd (Cover.index x✝.unop Q) i) ≫
      Multiequalizer.ι (Cover.index x✝.unop Q) i =
    0 ≫ Multiequalizer.ι (Cover.index x✝.unop Q) i
case w.h
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
x✝ : (Cover J X)ᵒᵖ
i : (Cover.index x✝.unop Q).L
⊢ (diagramNatTrans J 0 X).app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop Q) i =
    0.app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop Q) i
dsimp
no goals
case w.h
C : Type u
inst✝³ : Category C
J : GrothendieckTopology C
D : Type w
inst✝² : Category D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
P✝ : Cᵒᵖ ⥤ D
inst✝ : Preadditive D
X : C
P Q : Cᵒᵖ ⥤ D
x✝ : (Cover J X)ᵒᵖ
i : (Cover.index x✝.unop Q).L
⊢ Multiequalizer.lift (Cover.index x✝.unop Q) (multiequalizer (Cover.index x✝.unop P))
        (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ 0)
        (_ :
          ∀ (i : (Cover.index x✝.unop Q).R),
            (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ 0.app i.Y.op)
                  (MulticospanIndex.fstTo (Cover.index x✝.unop Q) i) ≫
                MulticospanIndex.fst (Cover.index x✝.unop Q) i =
              (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ 0.app i.Y.op)
                  (MulticospanIndex.sndTo (Cover.index x✝.unop Q) i) ≫
                MulticospanIndex.snd (Cover.index x✝.unop Q) i) ≫
      Multiequalizer.ι (Cover.index x✝.unop Q) i =
    0 ≫ Multiequalizer.ι (Cover.index x✝.unop Q) i
rw [zero_comp, Multiequalizer.lift_ι, comp_zero]