Module: Mathlib.CategoryTheory.Localization.SmallHom

Goal state before:
pp: 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 ≪≫
      Functor.isoWhiskerRight β.symm E₂ ≪≫
        Φ.functor.associator W₂.Q E₂ ≪≫
          Φ.functor.isoWhiskerLeft α₂ ≪≫ e ≪≫ Functor.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, ← NatIso.naturality_1 γ, ← Functor.comp_map,
    ← cancel_epi (e.inv.app X), ← cancel_epi (G.map (α₁.hom.app X)),
    ← cancel_epi (γ.hom.app (W₁.Q.obj X)), assoc, Iso.inv_hom_id_app_assoc,
    ← Functor.map_comp_assoc, Iso.hom_inv_id_app, Functor.map_id, id_comp,
    Iso.hom_inv_id_app_assoc]

Goal state after:
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 ≪≫
      Functor.isoWhiskerRight β.symm E₂ ≪≫
        Φ.functor.associator W₂.Q E₂ ≪≫
          Φ.functor.isoWhiskerLeft α₂ ≪≫ e ≪≫ Functor.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)
⊢ γ.hom.app (W₁.Q.obj X) ≫
      G.map (α₁.hom.app X) ≫
        e.inv.app X ≫
          α₂.inv.app (Φ.functor.obj X) ≫
            E₂.map (β.hom.app X) ≫ (G' ⋙ E₂).map g ≫ E₂.map (β.inv.app Y) ≫ α₂.hom.app (Φ.functor.obj Y) =
    (G' ⋙ E₂).map g ≫ γ.hom.app (W₁.Q.obj Y) ≫ G.map (α₁.hom.app Y) ≫ e.inv.app Y