-- 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
slice_lhs :
  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 ≫
              CategoryStruct.comp (Y :=
                tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y)
                  (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) Z X)))
                ((((obj L' X ◁ LaxMonoidal.μ L' Y Z ≫
                        CategoryStruct.comp (Y :=
                          obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z)))
                          (LaxMonoidal.μ L' X (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z))
                          (Functor.map L' (Iso.hom (β_ X (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z))) ≫
                            OplaxMonoidal.δ L' (tensorObj.{v_1, u_1} (𝒞 := inst) Y Z) 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 :=
        tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y)
          (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X 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 (β_ X Y) ▷ 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) Z X))
          (LaxMonoidal.μ L' X Z ≫ Functor.map L' (Iso.hom (β_ X Z))) (OplaxMonoidal.δ L' Z X))
⊢ 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 :=
        tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' Y)
          (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X 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 (β_ X Y) ▷ 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) Z X))
          (LaxMonoidal.μ L' X Z ≫ Functor.map L' (Iso.hom (β_ X Z))) (OplaxMonoidal.δ L' Z X))

Tactic:
slice_lhs 6 7 =>
    rw [braidingNatIso_hom_app_naturality_μ_left, braidingNatIso_hom_app]

Premises used: CategoryTheory.Localization.Monoidal.braidingNatIso_hom_app, CategoryTheory.Localization.Monoidal.braidingNatIso_hom_app_naturality_μ_left