-- Mathlib/CategoryTheory/Localization/SmallHom.lean
lemma equiv_smallHomMap (G : D₁ ⥤ D₂) (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G)
    (f : SmallHom.{w} W₁ X Y) :
    (SmallHom.equiv W₂ L₂) (Φ.smallHomMap f) =
      e.hom.app X ≫ G.map (SmallHom.equiv W₁ L₁ f) ≫ e.inv.app Y

Proof state:
case intro
C₁ : Type u₁
inst✝⁷ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁶ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
D₁ : Type u₃
inst✝⁵ : Category.{v₃, u₃} D₁
D₂ : Type u₄
inst✝⁴ : Category.{v₄, u₄} D₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
g : W₁.Q.obj X ⟶ W₁.Q.obj Y
G' : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
β : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ G' := CatCommSq.iso Φ.functor W₁.Q W₂.Q G'
E₁ : W₁.Localization ⥤ D₁ := (uniq W₁.Q L₁ W₁).functor
α₁ : W₁.Q ⋙ E₁ ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
E₂ : W₂.Localization ⥤ D₂ := (uniq W₂.Q L₂ W₂).functor
α₂ : W₂.Q ⋙ E₂ ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
γ : G' ⋙ E₂ ≅ E₁ ⋙ G :=
  liftNatIso W₁.Q W₁ (W₁.Q ⋙ G' ⋙ E₂) (W₁.Q ⋙ E₁ ⋙ G) (G' ⋙ E₂) (E₁ ⋙ G)
    ((W₁.Q.associator G' E₂).symm ≪≫
      isoWhiskerRight β.symm E₂ ≪≫
        Φ.functor.associator W₂.Q E₂ ≪≫
          isoWhiskerLeft Φ.functor α₂ ≪≫ e ≪≫ isoWhiskerRight α₁.symm G ≪≫ W₁.Q.associator E₁ G)
hγ :
  ∀ (X : C₁),
    γ.hom.app (W₁.Q.obj X) = E₂.map (β.inv.app X) ≫ α₂.hom.app (Φ.functor.obj X) ≫ e.hom.app X ≫ G.map (α₁.inv.app X)
⊢ α₂.inv.app (Φ.functor.obj X) ≫ E₂.map (β.hom.app X ≫ G'.map g ≫ β.inv.app Y) ≫ α₂.hom.app (Φ.functor.obj Y) =
    e.hom.app X ≫ G.map (α₁.inv.app X ≫ E₁.map g ≫ α₁.hom.app Y) ≫ e.inv.app Y

Tactic:
simp only [Functor.map_comp, assoc]

Result:
case intro
C₁ : Type u₁
inst✝⁷ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁶ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
D₁ : Type u₃
inst✝⁵ : Category.{v₃, u₃} D₁
D₂ : Type u₄
inst✝⁴ : Category.{v₄, u₄} D₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
g : W₁.Q.obj X ⟶ W₁.Q.obj Y
G' : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
β : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ G' := CatCommSq.iso Φ.functor W₁.Q W₂.Q G'
E₁ : W₁.Localization ⥤ D₁ := (uniq W₁.Q L₁ W₁).functor
α₁ : W₁.Q ⋙ E₁ ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
E₂ : W₂.Localization ⥤ D₂ := (uniq W₂.Q L₂ W₂).functor
α₂ : W₂.Q ⋙ E₂ ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
γ : G' ⋙ E₂ ≅ E₁ ⋙ G :=
  liftNatIso W₁.Q W₁ (W₁.Q ⋙ G' ⋙ E₂) (W₁.Q ⋙ E₁ ⋙ G) (G' ⋙ E₂) (E₁ ⋙ G)
    ((W₁.Q.associator G' E₂).symm ≪≫
      isoWhiskerRight β.symm E₂ ≪≫
        Φ.functor.associator W₂.Q E₂ ≪≫
          isoWhiskerLeft Φ.functor α₂ ≪≫ e ≪≫ isoWhiskerRight α₁.symm G ≪≫ W₁.Q.associator E₁ G)
hγ :
  ∀ (X : C₁),
    γ.hom.app (W₁.Q.obj X) = E₂.map (β.inv.app X) ≫ α₂.hom.app (Φ.functor.obj X) ≫ e.hom.app X ≫ G.map (α₁.inv.app X)
⊢ α₂.inv.app (Φ.functor.obj X) ≫
      E₂.map (β.hom.app X) ≫ E₂.map (G'.map g) ≫ E₂.map (β.inv.app Y) ≫ α₂.hom.app (Φ.functor.obj Y) =
    e.hom.app X ≫ G.map (α₁.inv.app X) ≫ G.map (E₁.map g) ≫ G.map (α₁.hom.app Y) ≫ e.inv.app Y

Proof state:
case intro
C₁ : Type u₁
inst✝⁷ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁶ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
D₁ : Type u₃
inst✝⁵ : Category.{v₃, u₃} D₁
D₂ : Type u₄
inst✝⁴ : Category.{v₄, u₄} D₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
g : W₁.Q.obj X ⟶ W₁.Q.obj Y
⊢ (SmallHom.equiv W₂ L₂) (Φ.smallHomMap ((SmallHom.equiv W₁ W₁.Q).symm g)) =
    e.hom.app X ≫ G.map ((SmallHom.equiv W₁ L₁) ((SmallHom.equiv W₁ W₁.Q).symm g)) ≫ e.inv.app Y

Tactic:
simp only [smallHomMap, Equiv.apply_symm_apply]

Result:
case intro
C₁ : Type u₁
inst✝⁷ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁶ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
D₁ : Type u₃
inst✝⁵ : Category.{v₃, u₃} D₁
D₂ : Type u₄
inst✝⁴ : Category.{v₄, u₄} D₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
g : W₁.Q.obj X ⟶ W₁.Q.obj Y
⊢ (SmallHom.equiv W₂ L₂)
      ((SmallHom.equiv W₂ W₂.Q).symm
        ((((CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).symm.app X).homCongr
            ((CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).symm.app Y))
          ((Φ.localizedFunctor W₁.Q W₂.Q).map g))) =
    e.hom.app X ≫ G.map ((SmallHom.equiv W₁ L₁) ((SmallHom.equiv W₁ W₁.Q).symm g)) ≫ e.inv.app Y