-- Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
theorem sheafifyCompIso_inv_eq_sheafifyLift :
    (J.sheafifyCompIso F P).inv =
      J.sheafifyLift (whiskerRight (J.toSheafify P) F)
        (HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _)))

Proof state:
C : Type u
inst✝¹³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹² : Category.{max v u, w₁} D
E : Type w₂
inst✝¹¹ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝¹⁰ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁹ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁵ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝³ : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
⊢ (J.sheafifyCompIso F P).inv = J.sheafifyLift (whiskerRight (J.toSheafify P) F) ⋯

Tactic:
apply J.sheafifyLift_unique

Result:
case a
C : Type u
inst✝¹³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹² : Category.{max v u, w₁} D
E : Type w₂
inst✝¹¹ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝¹⁰ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁹ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁵ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝³ : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
⊢ J.toSheafify (P ⋙ F) ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify P) F

Proof state:
case a
C : Type u
inst✝¹³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹² : Category.{max v u, w₁} D
E : Type w₂
inst✝¹¹ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝¹⁰ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁹ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁵ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝³ : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
⊢ J.toSheafify (P ⋙ F) = whiskerRight (J.toSheafify P) F ≫ (J.sheafifyCompIso F P).hom

Tactic:
simp

Result:
no goals

Proof state:
case a
C : Type u
inst✝¹³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹² : Category.{max v u, w₁} D
E : Type w₂
inst✝¹¹ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝¹⁰ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁹ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁵ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝³ : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
⊢ J.toSheafify (P ⋙ F) ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify P) F

Tactic:
rw [Iso.comp_inv_eq]

Result:
case a
C : Type u
inst✝¹³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹² : Category.{max v u, w₁} D
E : Type w₂
inst✝¹¹ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝¹⁰ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁹ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝⁸ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁵ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝³ : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
⊢ J.toSheafify (P ⋙ F) = whiskerRight (J.toSheafify P) F ≫ (J.sheafifyCompIso F P).hom