[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P✝ P : Cᵒᵖ ⥤ D
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ (whiskeringLeft Cᵒᵖ D E).obj (sheafify J P) ≅ (whiskeringLeft Cᵒᵖ D E).obj P ⋙ sheafification J E
[PROOFSTEP]
refine' J.plusFunctorWhiskerLeftIso _ ≪≫ _ ≪≫ Functor.associator _ _ _
[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P✝ P : Cᵒᵖ ⥤ D
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ (whiskeringLeft Cᵒᵖ D E).obj (plusObj J P) ⋙ plusFunctor J E ≅
    ((whiskeringLeft Cᵒᵖ D E).obj P ⋙ plusFunctor J E) ⋙ plusFunctor J E
[PROOFSTEP]
refine' isoWhiskerRight _ _
[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P✝ P : Cᵒᵖ ⥤ D
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ (whiskeringLeft Cᵒᵖ D E).obj (plusObj J P) ≅ (whiskeringLeft Cᵒᵖ D E).obj P ⋙ plusFunctor J E
[PROOFSTEP]
refine' J.plusFunctorWhiskerLeftIso _
[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝
P✝ P : Cᵒᵖ ⥤ D
F : D ⥤ E
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ NatTrans.app (sheafificationWhiskerLeftIso J P).hom F = (sheafifyCompIso J F P).hom
[PROOFSTEP]
dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]
[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝
P✝ P : Cᵒᵖ ⥤ D
F : D ⥤ E
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom ≫ 𝟙 (plusObj J (plusObj J (P ⋙ F))) =
    (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom
[PROOFSTEP]
rw [Category.comp_id]
[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝
P✝ P : Cᵒᵖ ⥤ D
F : D ⥤ E
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ NatTrans.app (sheafificationWhiskerLeftIso J P).inv F = (sheafifyCompIso J F P).inv
[PROOFSTEP]
dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]
[GOAL]
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝
inst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝
P✝ P : Cᵒᵖ ⥤ D
F : D ⥤ E
inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝ :
  (F : D ⥤ E) →
    (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
⊢ (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
[PROOFSTEP]
erw [Category.id_comp]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ sheafification J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ≅ (whiskeringRight Cᵒᵖ D E).obj F ⋙ sheafification J E
[PROOFSTEP]
refine' Functor.associator _ _ _ ≪≫ _
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ plusFunctor J D ⋙ plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ≅
    (whiskeringRight Cᵒᵖ D E).obj F ⋙ sheafification J E
[PROOFSTEP]
refine' isoWhiskerLeft (J.plusFunctor D) (J.plusFunctorWhiskerRightIso _) ≪≫ _
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E ≅
    (whiskeringRight Cᵒᵖ D E).obj F ⋙ sheafification J E
[PROOFSTEP]
refine' _ ≪≫ Functor.associator _ _ _
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E ≅
    ((whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E) ⋙ plusFunctor J E
[PROOFSTEP]
refine' (Functor.associator _ _ _).symm ≪≫ _
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ (plusFunctor J D ⋙ (whiskeringRight Cᵒᵖ D E).obj F) ⋙ plusFunctor J E ≅
    ((whiskeringRight Cᵒᵖ D E).obj F ⋙ plusFunctor J E) ⋙ plusFunctor J E
[PROOFSTEP]
exact isoWhiskerRight (J.plusFunctorWhiskerRightIso _) (J.plusFunctor E)
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ NatTrans.app (sheafificationWhiskerRightIso J F).hom P = (sheafifyCompIso J F P).hom
[PROOFSTEP]
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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) ≫
      (plusCompIso J F (plusObj J P)).hom ≫
        (𝟙 (plusObj J (plusObj J P ⋙ F)) ≫ plusMap J (plusCompIso J F P).hom) ≫ 𝟙 (plusObj J (plusObj J (P ⋙ F))) =
    (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom
[PROOFSTEP]
simp only [Category.id_comp, Category.comp_id]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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) ≫ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =
    (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom
[PROOFSTEP]
erw [Category.id_comp]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ NatTrans.app (sheafificationWhiskerRightIso J F).inv P = (sheafifyCompIso J F P).inv
[PROOFSTEP]
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
[PROOFSTEP]
simp only [Category.id_comp, Category.comp_id]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
[PROOFSTEP]
erw [Category.id_comp]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom = toSheafify J (P ⋙ F)
[PROOFSTEP]
dsimp [sheafifyCompIso]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ whiskerRight (toSheafify J P) F ≫ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =
    toSheafify J (P ⋙ F)
[PROOFSTEP]
erw [whiskerRight_comp, Category.assoc]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ whiskerRight (toPlus J P) F ≫
      whiskerRight (plusMap J (toPlus J P)) F ≫
        (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =
    toSheafify J (P ⋙ F)
[PROOFSTEP]
slice_lhs 2 3 => rw [plusCompIso_whiskerRight]
[GOAL]
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| whiskerRight (plusMap J (toPlus J P)) F ≫ (plusCompIso J F (plusObj J P)).hom
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| plusMap J (plusCompIso J F P).hom
case a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| whiskerRight (toPlus J P) F
[PROOFSTEP]
rw [plusCompIso_whiskerRight]
[GOAL]
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| whiskerRight (plusMap J (toPlus J P)) F ≫ (plusCompIso J F (plusObj J P)).hom
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| plusMap J (plusCompIso J F P).hom
case a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| whiskerRight (toPlus J P) F
[PROOFSTEP]
rw [plusCompIso_whiskerRight]
[GOAL]
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| whiskerRight (plusMap J (toPlus J P)) F ≫ (plusCompIso J F (plusObj J P)).hom
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| plusMap J (plusCompIso J F P).hom
case a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
| whiskerRight (toPlus J P) F
[PROOFSTEP]
rw [plusCompIso_whiskerRight]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ whiskerRight (toPlus J P) F ≫
      ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight (toPlus J P) F)) ≫ plusMap J (plusCompIso J F P).hom =
    toSheafify J (P ⋙ F)
[PROOFSTEP]
rw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ← Category.assoc,
  whiskerRight_toPlus_comp_plusCompIso_hom]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
⊢ toPlus J (P ⋙ F) ≫ plusMap J (toPlus J (P ⋙ F)) = toSheafify J (P ⋙ F)
[PROOFSTEP]
rfl
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
[PROOFSTEP]
rw [Iso.comp_inv_eq]
[GOAL]
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} 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
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
inst✝ : ReflectsIsomorphisms (forget D)
⊢ (sheafifyCompIso J F P).inv =
    sheafifyLift J (whiskerRight (toSheafify J P) F) (_ : Presheaf.IsSheaf J (sheafify J P ⋙ F))
[PROOFSTEP]
apply J.sheafifyLift_unique
[GOAL]
case a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
inst✝ : ReflectsIsomorphisms (forget D)
⊢ toSheafify J (P ⋙ F) ≫ (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F
[PROOFSTEP]
rw [Iso.comp_inv_eq]
[GOAL]
case a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
inst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P : Cᵒᵖ ⥤ D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
inst✝ : ReflectsIsomorphisms (forget D)
⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom
[PROOFSTEP]
simp