-- Module: Mathlib.CategoryTheory.Localization.Monoidal.Basic
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
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)))
⊢ Eq (α :=
    Quiver.Hom (V := LocalizedMonoidal L W ε)
      (Functor.obj L'
        (tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂) X₃)
          X₄))
      (Functor.obj L'
        (tensorObj.{v_1, u_1} (𝒞 := inst) (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂)
          (tensorObj.{v_1, u_1} (𝒞 := inst) X₃ X₄))))
    (Functor.map L' (Iso.hom (α_ (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂) X₃ X₄)))
    ((((((Functor.map L' (Iso.hom (α_ (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂) X₃ X₄)) ≫
                Iso.inv (μ L W ε (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂) (tensorObj.{v_1, u_1} (𝒞 := inst) X₃ X₄))) ≫
              Functor.obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂) ◁ Iso.inv (μ L W ε X₃ X₄)) ≫
            Iso.inv (μ L W ε X₁ X₂) ▷
              tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₃) (Functor.obj L' X₄)) ≫
          tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) (Functor.obj L' X₁) (Functor.obj L' X₂) ◁
            Iso.hom (μ L W ε X₃ X₄)) ≫
        Iso.hom (μ L W ε X₁ X₂) ▷ Functor.obj L' (tensorObj.{v_1, u_1} (𝒞 := inst) X₃ X₄)) ≫
      Iso.hom (μ L W ε (tensorObj.{v_1, u_1} (𝒞 := inst) X₁ X₂) (tensorObj.{v_1, u_1} (𝒞 := inst) X₃ X₄)))

Tactic:
simp only [← cancel_mono (μ L W ε (X₁ ⊗ X₂) (X₃ ⊗ X₄)).inv, assoc, id_comp,
    whisker_exchange_assoc, ← whiskerRight_comp_assoc,
    Iso.inv_hom_id, whiskerRight_id, ← whiskerLeft_comp,
    whiskerLeft_id]

Premises used: CategoryTheory.Localization.Monoidal.whiskerLeft_comp, CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id, CategoryTheory.cancel_mono, CategoryTheory.Localization.Monoidal.whiskerRight_id, CategoryTheory.Localization.Monoidal.whisker_exchange_assoc, CategoryTheory.Localization.Monoidal.whiskerRight_comp_assoc, CategoryTheory.MonoidalCategoryStruct.tensorObj, CategoryTheory.Localization.Monoidal.whiskerLeft_id, CategoryTheory.Localization.Monoidal.μ, CategoryTheory.Iso.inv, CategoryTheory.Category.id_comp