-- 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) ⟶ obj.{v₃, v₁, u₃, u₁} (obj (diagram V F₁ F₂) (Opposite.op i)) j)
    (end_.π (diagram V F₁ F₂) i ≫ map (obj (diagram V F₁ F₂) (Opposite.op i)) f)
    (end_.π (diagram V F₁ F₂) j ≫ NatTrans.app (map (diagram V F₁ F₂) f.op) j)
dsimp :
  Eq.{v₁ + 1} (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (enrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₁ F₂)
        (enrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₂ F₃) ⟶
      @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (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))))
    (CategoryStruct.comp (Y := @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj j) (F₃.obj j))
      ((end_.π (diagram V F₁ F₂) j ⊗ₘ end_.π (diagram V F₂ F₃) 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)))))
⊢ Eq.{v₁ + 1} (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (enrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₁ F₂)
        (enrichedHom.{v₁, v₂, v₃, u₁, u₂, u₃} V F₂ F₃) ⟶
      @EnrichedCategory.Hom.{v₁, u₁, u₂} _ inst inst_1 _ _ (F₁.obj i) (F₃.obj j))
    ((end_.π (diagram V F₁ F₂) i ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫
      Iso.inv (ρ_ (obj.{v₃, v₁, u₃, u₁} (obj (diagram V F₁ F₂) (Opposite.op i)) i)) ▷
          obj.{v₃, v₁, u₃, u₁} (obj (diagram V F₂ F₃) (Opposite.op j)) j ≫
        (obj.{v₃, v₁, u₃, u₁} (obj (diagram V F₁ F₂) (Opposite.op i)) i ◁
              (eHomEquiv V : (a : F₂.obj i ⟶ F₂.obj j) → 𝟙_ V ⟶ F₂.obj i ⟶[V] F₂.obj j) (F₂.map f)) ▷
            obj.{v₃, v₁, u₃, u₁} (obj (diagram V F₂ F₃) (Opposite.op j)) j ≫
          eComp V (Opposite.unop (F₁.op.obj (Opposite.op i))) (F₂.obj i) (F₂.obj j) ▷
              obj.{v₃, v₁, u₃, u₁} (obj (diagram V F₂ F₃) (Opposite.op j)) j ≫
            eComp V (Opposite.unop (F₁.op.obj (Opposite.op i))) (F₂.obj j) (F₃.obj j))
    (((end_.π (diagram V F₁ F₂) j ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫ eComp V (F₁.obj j) (F₂.obj j) (F₃.obj j)) ≫
      eHomWhiskerRight V (F₁.map f) (F₃.obj j))

Tactic:
dsimp [eHomWhiskerLeft, eHomWhiskerRight] at this ⊢

Premises used: CategoryTheory.eHomWhiskerLeft, CategoryTheory.eHomWhiskerRight