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
⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom
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
⊢ toSheafify J (P ⋙ F) ≫ (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F
rw [Iso.comp_inv_eq]
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
⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom
simp