-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.Monoidal.Basic (hypothesis)
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
Y₁ Y₂ Y₃ Y₄ : LocalizedMonoidal L W ε
X₁ : C
e₁ : Functor.obj L' X₁ ≅ Y₁
X₂ : C
e₂ : Functor.obj L' X₂ ≅ Y₂
X₃ : C
e₃ : Functor.obj L' X₃ ≅ Y₃
X₄ : C
e₄ : Functor.obj L' X₄ ≅ Y₄
⊢ 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)
          (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₁) (Functor.obj L' X₂))
          (Functor.obj L' X₃))
        (Functor.obj L' X₄))
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₁)
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₂)
          (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₃) (Functor.obj L' X₄)))))
    (Iso.hom (α_ (Functor.obj L' X₁) (Functor.obj L' X₂) (Functor.obj L' X₃)) ▷ Functor.obj L' X₄ ≫
      Iso.hom
          (α_ (Functor.obj L' X₁)
            (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₂) (Functor.obj L' X₃))
            (Functor.obj L' X₄)) ≫
        Functor.obj L' X₁ ◁ Iso.hom (α_ (Functor.obj L' X₂) (Functor.obj L' X₃) (Functor.obj L' X₄)))
    (Iso.hom
        (α_ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₁) (Functor.obj L' X₂))
          (Functor.obj L' X₃) (Functor.obj L' X₄)) ≫
      Iso.hom
        (α_ (Functor.obj L' X₁) (Functor.obj L' X₂)
          (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₃) (Functor.obj L' X₄))))
have this : 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) (Functor.obj L' X₁)
        (Functor.obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X₂ X₃)))
      (Functor.obj L' X₄))
    (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₁)
      (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε)
        (Functor.obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X₂ X₃)) (Functor.obj L' X₄))))
  ((Functor.obj L' X₁ ◁ Iso.inv (μ L W ε X₂ X₃)) ▷ Functor.obj L' X₄ ≫
    Iso.hom
        (α_ (Functor.obj L' X₁)
          (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₂) (Functor.obj L' X₃))
          (Functor.obj L' X₄)) ≫
      Functor.obj L' X₁ ◁ Iso.hom (μ L W ε X₂ X₃) ▷ Functor.obj L' X₄)
  (Iso.hom (α_ (Functor.obj L' X₁) (Functor.obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X₂ X₃)) (Functor.obj L' X₄)))
Tactic: have : ((L').obj X₁ ◁ (μ L W ε X₂ X₃).inv) ▷ (L').obj X₄ ≫
      (α_ ((L').obj X₁) ((L').obj X₂ ⊗ (L').obj X₃) ((L').obj X₄)).hom ≫
        (L').obj X₁ ◁ (μ L W ε X₂ X₃).hom ▷ (L').obj X₄ =
          (α_ ((L').obj X₁) ((L').obj (X₂ ⊗ X₃)) ((L').obj X₄)).hom :=
    pentagon_aux₂ _ _ _ (μ L W ε X₂ X₃).symm