-- Module: Mathlib.CategoryTheory.Localization.Monoidal.Braided
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : W.IsMonoidal
inst_4 : L.IsLocalization W
unit : D
ε : L.obj (tensorUnit.{v_1, u_1} C (𝒞 := inst)) ≅ unit
inst_5 : BraidedCategory.{v_1, u_1} C
X Y Z : C
simp :
  Eq (α :=
    Quiver.Hom (V := LocalizedMonoidal L W ε)
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y) (obj L' Z)))
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Z) (obj L' X)) (obj L' Y)))
    (CategoryStruct.comp (Y :=
        tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X)
          (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z)))
        (obj L' X ◁ LaxMonoidal.μ L' Y Z)
        (LaxMonoidal.μ L' X (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z) ≫
          Functor.map L' (Iso.inv (α_ X Y Z)) ≫
            OplaxMonoidal.δ L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) Z ≫ OplaxMonoidal.δ L' X Y ▷ obj L' Z) ≫
      NatTrans.app
          (NatTrans.app (Iso.hom (braidingNatIso L W ε))
            (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X) (obj L' Y)))
          (obj L' Z) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Z)
            (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Y)))
          (obj L' Z ◁ LaxMonoidal.μ L' X Y)
          (LaxMonoidal.μ L' Z (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ≫
            Functor.map L' (Iso.inv (α_ Z X Y)) ≫
              OplaxMonoidal.δ L' (tensorObj.{v_1, u_1} (𝒞 := inst) Z X) Y ≫ OplaxMonoidal.δ L' Z X ▷ obj L' Y))
    (obj L' X ◁
        CategoryStruct.comp (Y := obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z)) (LaxMonoidal.μ L' Y Z)
          (Functor.map L' (Iso.hom (β_ Y Z)) ≫ OplaxMonoidal.δ L' Z Y) ≫
      CategoryStruct.comp (Y :=
          tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X)
            (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) Z Y)))
          (obj L' X ◁ LaxMonoidal.μ L' Z Y)
          (LaxMonoidal.μ L' X (tensorObj.{v_1, u_1} (𝒞 := inst) Z Y) ≫
            Functor.map L' (Iso.inv (α_ X Z Y)) ≫
              OplaxMonoidal.δ L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Z) Y ≫ OplaxMonoidal.δ L' X Z ▷ obj L' Y) ≫
        CategoryStruct.comp (Y := obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Z)) (LaxMonoidal.μ L' X Z)
            (Functor.map L' (Iso.hom (β_ X Z)) ≫ OplaxMonoidal.δ L' Z X) ▷
          obj L' Y)
⊢ Eq (α :=
    Quiver.Hom (V := LocalizedMonoidal L W ε)
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y) (obj L' Z)))
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Z) (obj L' X)) (obj L' Y)))
    (Iso.inv (α_ (obj L' X) (obj L' Y) (obj L' Z)) ≫
      Iso.hom
          (Iso.app
            (Iso.app (braidingNatIso L W ε)
              (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X) (obj L' Y)))
            (obj L' Z)) ≫
        Iso.inv (α_ (obj L' Z) (obj L' X) (obj L' Y)))
    (obj L' X ◁ Iso.hom (Iso.app (Iso.app (braidingNatIso L W ε) (obj L' Y)) (obj L' Z)) ≫
      Iso.inv (α_ (obj L' X) (obj L' Z) (obj L' Y)) ≫
        Iso.hom (Iso.app (Iso.app (braidingNatIso L W ε) (obj L' X)) (obj L' Z)) ▷ obj L' Y)

Tactic:
simp only [associator_inv, Iso.app_hom, braidingNatIso_hom_app]

Premises used: CategoryTheory.associator_inv, CategoryTheory.Iso.app_hom, CategoryTheory.Localization.Monoidal.braidingNatIso_hom_app