Theorem: CategoryTheory.Functor.IsLocalization.mk'
File: Mathlib/CategoryTheory/Localization/Predicate.lean

tactic: simp only [← Functor.assoc, Localization.Construction.fac, h₂.fac,
  Functor.comp_id]
state_before: C : Type u_1
D : Type u_2
inst✝² : Category C
inst✝¹ : Category D
L : C ⥤ D
W : MorphismProperty C
E : Type ?u.6483
inst✝ : Category E
h₁ : Localization.StrictUniversalPropertyFixedTarget L W D
h₂ : Localization.StrictUniversalPropertyFixedTarget L W (MorphismProperty.Localization W)
⊢ MorphismProperty.Q W ⋙ 𝟭 (MorphismProperty.Localization W) =
    MorphismProperty.Q W ⋙
      Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L) ⋙
        Localization.StrictUniversalPropertyFixedTarget.lift h₂ (MorphismProperty.Q W)
          (_ : MorphismProperty.IsInvertedBy W (MorphismProperty.Q W))

tactic: simp only [← Functor.assoc, h₂.fac, Localization.Construction.fac,
  Functor.comp_id]
state_before: C : Type u_1
D : Type u_2
inst✝² : Category C
inst✝¹ : Category D
L : C ⥤ D
W : MorphismProperty C
E : Type ?u.6483
inst✝ : Category E
h₁ : Localization.StrictUniversalPropertyFixedTarget L W D
h₂ : Localization.StrictUniversalPropertyFixedTarget L W (MorphismProperty.Localization W)
⊢ L ⋙
      Localization.StrictUniversalPropertyFixedTarget.lift h₂ (MorphismProperty.Q W)
          (_ : MorphismProperty.IsInvertedBy W (MorphismProperty.Q W)) ⋙
        Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L) =
    L ⋙ 𝟭 D

tactic: simp only [eqToIso.hom, eqToHom_app, eqToHom_map, eqToHom_trans,
  eqToHom_refl]
state_before: C : Type u_1
D : Type u_2
inst✝² : Category C
inst✝¹ : Category D
L : C ⥤ D
W : MorphismProperty C
E : Type ?u.6483
inst✝ : Category E
h₁ : Localization.StrictUniversalPropertyFixedTarget L W D
h₂ : Localization.StrictUniversalPropertyFixedTarget L W (MorphismProperty.Localization W)
X : MorphismProperty.Localization W
⊢ (Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)).map
        ((eqToIso
                (_ :
                  𝟭 (MorphismProperty.Localization W) =
                    Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L) ⋙
                      Localization.StrictUniversalPropertyFixedTarget.lift h₂ (MorphismProperty.Q W)
                        (_ : MorphismProperty.IsInvertedBy W (MorphismProperty.Q W)))).hom.app
          X) ≫
      (eqToIso
              (_ :
                Localization.StrictUniversalPropertyFixedTarget.lift h₂ (MorphismProperty.Q W)
                      (_ : MorphismProperty.IsInvertedBy W (MorphismProperty.Q W)) ⋙
                    Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L) =
                  𝟭 D)).hom.app
        ((Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)).obj X) =
    𝟙 ((Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)).obj X)

tactic: rfl
state_before: C : Type u_1
D : Type u_2
inst✝² : Category C
inst✝¹ : Category D
L : C ⥤ D
W : MorphismProperty C
E : Type ?u.6483
inst✝ : Category E
h₁ : Localization.StrictUniversalPropertyFixedTarget L W D
h₂ : Localization.StrictUniversalPropertyFixedTarget L W (MorphismProperty.Localization W)
X : MorphismProperty.Localization W
⊢ 𝟙
      ((Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)).obj
        ((𝟭 (MorphismProperty.Localization W)).obj X)) =
    𝟙 ((Localization.Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)).obj X)