Theorem: CategoryTheory.LocalizerMorphism.isRightDerivabilityStructure_iff
File: Mathlib/CategoryTheory/Localization/DerivabilityStructure/Basic.lean

Proof steps:
state_before: 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
⊢ Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom
state_after: 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
⊢ Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom
tactic: have : Φ.IsRightDerivabilityStructure ↔
      TwoSquare.GuitartExact ((Φ.catCommSq W₁.Q W₂.Q).iso).hom :=
    ⟨fun h => h.guitartExact', fun h => ⟨inferInstance, h⟩⟩
state_before: 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
⊢ Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom
state_after: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: rw [this]
state_before: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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)
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let e' := (Φ.catCommSq W₁.Q W₂.Q).iso
state_before: 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)
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let E₁ := Localization.uniq W₁.Q L₁ W₁
state_before: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let E₂ := Localization.uniq W₂.Q L₂ W₂
state_before: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
state_before: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
state_before: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
    Functor.associator _ _ _ ≪≫ isoWhiskerLeft _ e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
state_before: 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
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
tactic: let e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
    liftNatIso W₁.Q W₁ _ _ _ _ e''
state_before: 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.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
state_after: 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: have : TwoSquare.vComp' e'.hom e'''.hom e₁ e₂ = e.hom := by
    ext X₁
    rw [TwoSquare.vComp'_app, liftNatIso_hom, liftNatTrans_app]
    simp only [Functor.comp_obj, Iso.trans_hom, isoWhiskerLeft_hom, isoWhiskerRight_hom,
      Iso.symm_hom, NatTrans.comp_app, Functor.associator_hom_app, whiskerLeft_app,
      whiskerRight_app, id_comp, assoc, e'']
    dsimp [Lifting.iso]
    rw [F.map_id, id_comp, ← F.map_comp, Iso.inv_hom_id_app, F.map_id, comp_id,
      ← Functor.map_comp_assoc]
    erw [show (CatCommSq.iso Φ.functor W₁.Q W₂.Q (localizedFunctor Φ W₁.Q W₂.Q)).hom =
      (Lifting.iso W₁.Q W₁ _ _).inv by rfl, Iso.inv_hom_id_app]
    simp
state_before: 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
state_after: no goals
tactic: rw [← TwoSquare.GuitartExact.vComp'_iff_of_equivalences e'.hom E₁ E₂ e''' e₁ e₂, this]