-- Mathlib/CategoryTheory/Localization/DerivabilityStructure/Basic.lean
lemma isRightDerivabilityStructure_iff [Φ.HasRightResolutions] (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F) :
    Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom

Proof state:
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
e' : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ Φ.localizedFunctor W₁.Q W₂.Q :=
  CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
E₁ : W₁.Localization ≌ D₁ := uniq W₁.Q L₁ W₁
E₂ : W₂.Localization ≌ D₂ := uniq W₂.Q L₂ W₂
e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
  Φ.functor.associator W₂.Q E₂.functor ≪≫ isoWhiskerLeft Φ.functor e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
  liftNatIso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) ((W₁.Q ⋙ E₁.functor) ⋙ F)
    (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor) (E₁.functor ⋙ F) e''
⊢ TwoSquare.vComp' e'.hom e'''.hom e₁ e₂ = e.hom

Tactic:
ext X₁

Result:
case h
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
e' : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ Φ.localizedFunctor W₁.Q W₂.Q :=
  CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
E₁ : W₁.Localization ≌ D₁ := uniq W₁.Q L₁ W₁
E₂ : W₂.Localization ≌ D₂ := uniq W₂.Q L₂ W₂
e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
  Φ.functor.associator W₂.Q E₂.functor ≪≫ isoWhiskerLeft Φ.functor e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
  liftNatIso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) ((W₁.Q ⋙ E₁.functor) ⋙ F)
    (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor) (E₁.functor ⋙ F) e''
X₁ : C₁
⊢ (TwoSquare.vComp' e'.hom e'''.hom e₁ e₂).natTrans.app X₁ = (TwoSquare.natTrans e.hom).app X₁

Proof state:
case h
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
e' : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ Φ.localizedFunctor W₁.Q W₂.Q :=
  CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
E₁ : W₁.Localization ≌ D₁ := uniq W₁.Q L₁ W₁
E₂ : W₂.Localization ≌ D₂ := uniq W₂.Q L₂ W₂
e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
  Φ.functor.associator W₂.Q E₂.functor ≪≫ isoWhiskerLeft Φ.functor e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
  liftNatIso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) ((W₁.Q ⋙ E₁.functor) ⋙ F)
    (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor) (E₁.functor ⋙ F) e''
X₁ : C₁
⊢ (TwoSquare.vComp' e'.hom e'''.hom e₁ e₂).natTrans.app X₁ = (TwoSquare.natTrans e.hom).app X₁

Tactic:
rw [TwoSquare.vComp'_app, liftNatIso_hom, liftNatTrans_app]

Result:
case h
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
e' : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ Φ.localizedFunctor W₁.Q W₂.Q :=
  CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
E₁ : W₁.Localization ≌ D₁ := uniq W₁.Q L₁ W₁
E₂ : W₂.Localization ≌ D₂ := uniq W₂.Q L₂ W₂
e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
  Φ.functor.associator W₂.Q E₂.functor ≪≫ isoWhiskerLeft Φ.functor e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
  liftNatIso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) ((W₁.Q ⋙ E₁.functor) ⋙ F)
    (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor) (E₁.functor ⋙ F) e''
X₁ : C₁
⊢ e₂.inv.app (Φ.functor.obj X₁) ≫
      E₂.functor.map ((TwoSquare.natTrans e'.hom).app X₁) ≫
        ((Lifting.iso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor)).hom.app
              X₁ ≫
            e''.hom.app X₁ ≫ (Lifting.iso W₁.Q W₁ ((W₁.Q ⋙ E₁.functor) ⋙ F) (E₁.functor ⋙ F)).inv.app X₁) ≫
          F.map (e₁.hom.app X₁) =
    (TwoSquare.natTrans e.hom).app X₁

Proof state:
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this✝ :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
e' : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ Φ.localizedFunctor W₁.Q W₂.Q :=
  CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
E₁ : W₁.Localization ≌ D₁ := uniq W₁.Q L₁ W₁
E₂ : W₂.Localization ≌ D₂ := uniq W₂.Q L₂ W₂
e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
  Φ.functor.associator W₂.Q E₂.functor ≪≫ isoWhiskerLeft Φ.functor e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
  liftNatIso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) ((W₁.Q ⋙ E₁.functor) ⋙ F)
    (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor) (E₁.functor ⋙ F) e''
this : TwoSquare.vComp' e'.hom e'''.hom e₁ e₂ = e.hom
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom

Tactic:
rw [← TwoSquare.GuitartExact.vComp'_iff_of_equivalences e'.hom E₁ E₂ e''' e₁ e₂, this]

Result:
no goals