-- 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) (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))
    (CategoryStruct.comp (Y :=
        tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Z))
          (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)
              (obj L' X ◁ Functor.map L' (Iso.hom (β_ Y Z)) ≫
                obj L' X ◁ OplaxMonoidal.δ L' 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) Z X))
          (LaxMonoidal.μ L' X Z ≫ Functor.map L' (Iso.hom (β_ X Z))) (OplaxMonoidal.δ L' Z X) ▷
        obj L' Y)
have slice_lhs : 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 ≫
            ((((LaxMonoidal.μ L' X Y ▷ obj L' Z ≫
                      CategoryStruct.comp (Y :=
                        obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) Z))
                        (LaxMonoidal.μ L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) Z)
                        (Functor.map L' (Iso.hom (β_ (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) Z)) ≫
                          OplaxMonoidal.δ L' Z (tensorObj.{v_1, u_1} (𝒞 := inst) 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))
  (CategoryStruct.comp (Y :=
      tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X Z))
        (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)
            (obj L' X ◁ Functor.map L' (Iso.hom (β_ Y Z)) ≫
              obj L' X ◁ OplaxMonoidal.δ L' 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) Z X))
        (LaxMonoidal.μ L' X Z ≫ Functor.map L' (Iso.hom (β_ X Z))) (OplaxMonoidal.δ L' Z X) ▷
      obj L' Y)
Tactic: slice_lhs 6 7 =>
    rw [braidingNatIso_hom_app_naturality_μ_right, braidingNatIso_hom_app]