-- Module: Mathlib.CategoryTheory.Enriched.FunctorCategory
Goal:
V : Type u₁
inst : Category.{v₁, u₁} V
inst_1 : MonoidalCategory.{v₁, u₁} V
C : Type u₂
inst_2 : Category.{v₂, u₂} C
J : Type u₃
inst_3 : Category.{v₃, u₃} J
K : Type u₄
inst_4 : Category.{v₄, u₄} K
inst_5 : EnrichedOrdinaryCategory.{v₁, v₂, u₂, u₁} V C
F₁ F₂ F₃ F₄ : Functor.{v₃, v₂, u₃, u₂} J C
inst_6 : HasEnrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₁ F₂
inst_7 : HasEnrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₂ F₃
inst_8 : HasEnrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₁ F₃
i j : J
f : i ⟶ j
this :
  Eq.{v₁ + 1} (α :=
    end_.{v₃, v₁, u₃, u₁} (diagram V F₁ F₂) ⟶
      @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj j))
    (end_.π (diagram V F₁ F₂) i ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i))
          (tensorUnit.{v₁, u₁} V (𝒞 := inst)))
        (Iso.inv (ρ_ (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i))))
        (CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i))
            (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj i) (F₂.obj j)))
          (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i) ◁
            (eHomEquiv V : (a : F₂.obj i ⟶ F₂.obj j) → 𝟙_ V ⟶ F₂.obj i ⟶[V] F₂.obj j) (F₂.map f))
          (eComp V (F₁.obj i) (F₂.obj i) (F₂.obj j))))
    (end_.π (diagram V F₁ F₂) j ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} V (𝒞 := inst))
          (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₂.obj j)))
        (Iso.inv (λ_ (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₂.obj j))))
        (CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₁.obj j))
            (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₂.obj j)))
          ((eHomEquiv V : (a : F₁.obj i ⟶ F₁.obj j) → 𝟙_ V ⟶ F₁.obj i ⟶[V] F₁.obj j) (F₁.map f) ▷
            @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₂.obj j))
          (eComp V (F₁.obj i) (F₁.obj j) (F₂.obj j))))
rw :
  enrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₁ F₂ ◁ end_.π (diagram V F₂ F₃) j ≫
    end_.π (diagram V F₁ F₂) j ▷ @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj j) (F₃.obj j) ≫
      eComp V (F₁.obj j) (F₂.obj j) (F₃.obj j) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} V (𝒞 := inst))
            (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₃.obj j)))
          (Iso.inv (λ_ (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₃.obj j))))
          (CategoryStruct.comp (Y :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₁.obj j))
              (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₃.obj j)))
            ((eHomEquiv V : (a : F₁.obj i ⟶ F₁.obj j) → 𝟙_ V ⟶ F₁.obj i ⟶[V] F₁.obj j) (F₁.map f) ▷
              @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₃.obj j))
            (eComp V (F₁.obj i) (F₁.obj j) (F₃.obj j)))
| (end_.π (diagram V F₁ F₂) i ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i))
          (tensorUnit.{v₁, u₁} V (𝒞 := inst)))
        (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj j) (F₃.obj j)))
      (Iso.inv (ρ_ (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i))) ▷
        @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj j) (F₃.obj j))
      (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i))
            (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj i) (F₂.obj j)))
          (@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj j) (F₃.obj j)))
        ((@EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₂.obj i) ◁
            (eHomEquiv V : (a : F₂.obj i ⟶ F₂.obj j) → 𝟙_ V ⟶ F₂.obj i ⟶[V] F₂.obj j) (F₂.map f)) ▷
          @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj j) (F₃.obj j))
        (eComp V (F₁.obj i) (F₂.obj i) (F₂.obj j) ▷
            @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₂.obj j) (F₃.obj j) ≫
          eComp V (F₁.obj i) (F₂.obj j) (F₃.obj j)))

Tactic:
rw [tensorHom_def'_assoc, ← comp_whiskerRight_assoc,
          ← comp_whiskerRight_assoc, ← comp_whiskerRight_assoc,
          assoc, assoc, this, comp_whiskerRight_assoc, comp_whiskerRight_assoc,
          comp_whiskerRight_assoc, leftUnitor_inv_whiskerRight_assoc,
          ← associator_inv_naturality_left_assoc, ← e_assoc',
          Iso.inv_hom_id_assoc, ← whisker_exchange_assoc, id_whiskerLeft_assoc,
          Iso.inv_hom_id_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.associator_inv_naturality_left_assoc, CategoryTheory.MonoidalCategory.leftUnitor_inv_whiskerRight_assoc, CategoryTheory.e_assoc', CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.MonoidalCategory.id_whiskerLeft_assoc, CategoryTheory.MonoidalCategory.whisker_exchange_assoc, CategoryTheory.MonoidalCategory.tensorHom_def'_assoc, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc