C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
⊢ ((𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv ≫ 𝟙 (plusObj J (plusObj J P ⋙ F))) ≫
        (plusCompIso J F (plusObj J P)).inv) ≫
      𝟙 (plusObj J (plusObj J P) ⋙ F) =
    plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
⊢ (sheafificationWhiskerRightIso J F).inv.app P = (sheafifyCompIso J F P).inv
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
⊢ (𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv) ≫ (plusCompIso J F (plusObj J P)).inv =
    plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
⊢ ((𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv ≫ 𝟙 (plusObj J (plusObj J P ⋙ F))) ≫
        (plusCompIso J F (plusObj J P)).inv) ≫
      𝟙 (plusObj J (plusObj J P) ⋙ F) =
    plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv
simp only [Category.id_comp, Category.comp_id]
no goals
C : Type u
inst✝⁸ : Category C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category D
E : Type w₂
inst✝⁶ : Category E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
⊢ (𝟙 (plusObj J (plusObj J (P ⋙ F))) ≫ plusMap J (plusCompIso J F P).inv) ≫ (plusCompIso J F (plusObj J P)).inv =
    plusMap J (plusCompIso J F P).inv ≫ (plusCompIso J F (plusObj J P)).inv
erw [Category.id_comp]