-- Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
    whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _

Proof state:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toPlus P) F ≫
      whiskerRight (J.plusMap (J.toPlus P)) F ≫
        (J.plusCompIso F (J.plusObj P)).hom ≫ J.plusMap (J.plusCompIso F P).hom =
    J.toSheafify (P ⋙ F)

Tactic:
slice_lhs 2 3 => rw [plusCompIso_whiskerRight]

Result:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toPlus P) F ≫
      ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerRight (J.toPlus P) F)) ≫ J.plusMap (J.plusCompIso F P).hom =
    J.toSheafify (P ⋙ F)

Proof state:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toPlus P) F ≫
      ((J.plusCompIso F P).hom ≫ J.plusMap (whiskerRight (J.toPlus P) F)) ≫ J.plusMap (J.plusCompIso F P).hom =
    J.toSheafify (P ⋙ F)

Tactic:
rw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ←
  Category.assoc, whiskerRight_toPlus_comp_plusCompIso_hom]

Result:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ J.toPlus (P ⋙ F) ≫ J.plusMap (J.toPlus (P ⋙ F)) = J.toSheafify (P ⋙ F)

Proof state:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toSheafify P) F ≫ (J.plusCompIso F (J.plusObj P)).hom ≫ J.plusMap (J.plusCompIso F P).hom =
    J.toSheafify (P ⋙ F)

Tactic:
erw [whiskerRight_comp, Category.assoc]

Result:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toPlus P) F ≫
      whiskerRight (J.plusMap (J.toPlus P)) F ≫
        (J.plusCompIso F (J.plusObj P)).hom ≫ J.plusMap (J.plusCompIso F P).hom =
    J.toSheafify (P ⋙ F)

Proof state:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ J.toPlus (P ⋙ F) ≫ J.plusMap (J.toPlus (P ⋙ F)) = J.toSheafify (P ⋙ F)

Tactic:
rfl

Result:
no goals

Proof state:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toSheafify P) F ≫ (J.sheafifyCompIso F P).hom = J.toSheafify (P ⋙ F)

Tactic:
dsimp [sheafifyCompIso]

Result:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝¹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
⊢ whiskerRight (J.toSheafify P) F ≫ (J.plusCompIso F (J.plusObj P)).hom ≫ J.plusMap (J.plusCompIso F P).hom =
    J.toSheafify (P ⋙ F)