-- 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 :
  Pentagon.{v_2, u_2} (C := LocalizedMonoidal L W ε) (Functor.obj L' X₁) (Functor.obj L' X₂) (Functor.obj L' X₃)
    (Functor.obj L' X₄)
refine_1 :
  Eq (α :=
    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) Y₁ Y₂) Y₃)
        Y₄ ⟶
      tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₁
        (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₂
          (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₃ Y₄)))
    (Iso.hom (α_ Y₁ Y₂ Y₃) ▷ Y₄ ≫
      Iso.hom (α_ Y₁ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₂ Y₃) Y₄) ≫
        Y₁ ◁ Iso.hom (α_ Y₂ Y₃ Y₄))
    ((((e₁.inv ⊗ₘ e₂.inv) ⊗ₘ e₃.inv) ⊗ₘ e₄.inv) ≫
      (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))) ≫
        (e₁.hom ⊗ₘ e₂.hom ⊗ₘ e₃.hom ⊗ₘ e₄.hom))
refine_2 :
  (((e₁.inv ⊗ₘ e₂.inv) ⊗ₘ e₃.inv) ⊗ₘ e₄.inv) ≫
      (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)))) ≫
        (e₁.hom ⊗ₘ e₂.hom ⊗ₘ e₃.hom ⊗ₘ e₄.hom) =
    Iso.hom (α_ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₁ Y₂) Y₃ Y₄) ≫
      Iso.hom (α_ Y₁ Y₂ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₃ Y₄))
⊢ Iso.hom (α_ Y₁ Y₂ Y₃) ▷ Y₄ ≫
      Iso.hom (α_ Y₁ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₂ Y₃) Y₄) ≫
        Y₁ ◁ Iso.hom (α_ Y₂ Y₃ Y₄) =
    Iso.hom (α_ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₁ Y₂) Y₃ Y₄) ≫
      Iso.hom (α_ Y₁ Y₂ (tensorObj.{v_2, u_2} (𝒞 := instCategoryLocalizedMonoidal L W ε) Y₃ Y₄))

Tactic:
refine Eq.trans ?_ (((((e₁.inv ⊗ₘ e₂.inv) ⊗ₘ e₃.inv) ⊗ₘ e₄.inv) ≫= this =≫
      (e₁.hom ⊗ₘ e₂.hom ⊗ₘ e₃.hom ⊗ₘ e₄.hom)).trans ?_)

Premises used: CategoryTheory.eq_whisker, CategoryTheory.MonoidalCategoryStruct.tensorHom, CategoryTheory.whisker_eq, CategoryTheory.Iso.hom, CategoryTheory.Iso.inv, Eq.trans