-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.Monoidal.Braided (goal)
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
⊢ Eq (α :=
    Quiver.Hom (V := LocalizedMonoidal L W ε)
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X) (obj L' Y)) (obj L' Z))
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Z) (obj L' X))))
    (Iso.hom (α_ (obj L' X) (obj L' Y) (obj L' Z)) ≫
      Iso.hom
          (Iso.app (Iso.app (braidingNatIso L W ε) (obj L' X))
            (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y) (obj L' Z))) ≫
        Iso.hom (α_ (obj L' Y) (obj L' Z) (obj L' X)))
    (Iso.hom (Iso.app (Iso.app (braidingNatIso L W ε) (obj L' X)) (obj L' Y)) ▷ obj L' Z ≫
      Iso.hom (α_ (obj L' Y) (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)))
have simp : Eq (α :=
  Quiver.Hom (V := LocalizedMonoidal L W ε)
    (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε)
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' X) (obj L' Y)) (obj L' Z))
    (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y)
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Z) (obj L' X))))
  (CategoryStruct.comp (Y :=
      tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Y))
        (obj L' Z))
      (LaxMonoidal.μ L' X Y ▷ obj L' Z)
      (LaxMonoidal.μ L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) Z ≫
        Functor.map L' (Iso.hom (α_ X Y Z)) ≫
          OplaxMonoidal.δ L' X (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z) ≫ obj L' X ◁ OplaxMonoidal.δ L' Y Z) ≫
    NatTrans.app (NatTrans.app (Iso.hom (braidingNatIso L W ε)) (obj L' X))
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y) (obj L' Z)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z))
          (obj L' X))
        (LaxMonoidal.μ L' Y Z ▷ obj L' X)
        (LaxMonoidal.μ L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z) X ≫
          Functor.map L' (Iso.hom (α_ Y Z X)) ≫
            OplaxMonoidal.δ L' Y (tensorObj.{v_1, u_1} (𝒞 := inst) Z X) ≫ obj L' Y ◁ OplaxMonoidal.δ L' Z X))
  (CategoryStruct.comp (Y := obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Y)) (LaxMonoidal.μ L' X Y)
        (Functor.map L' (Iso.hom (β_ X Y)) ≫ OplaxMonoidal.δ L' Y X) ▷
      obj L' Z ≫
    CategoryStruct.comp (Y :=
        tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y X))
          (obj L' Z))
        (LaxMonoidal.μ L' Y X ▷ obj L' Z)
        (LaxMonoidal.μ L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y X) Z ≫
          Functor.map L' (Iso.hom (α_ Y X Z)) ≫
            OplaxMonoidal.δ L' Y (tensorObj.{v_1, u_1} (𝒞 := inst) X Z) ≫ obj L' 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))
Tactic: simp only [associator_hom, Iso.app_hom, braidingNatIso_hom_app]