-- Module: Mathlib.CategoryTheory.Enriched.Ordinary.Basic
Goal:
V : Type u'
inst : Category.{v', u'} V
inst_1 : MonoidalCategory.{v', u'} V
C : Type u
inst_2 : Category.{v, u} C
inst_3 : EnrichedOrdinaryCategory.{v', v, u, u'} V C
W_1 : Type u''
inst_4 : Category.{v'', u''} W_1
inst_5 : MonoidalCategory.{v'', u''} W_1
F_1 : Functor.{v', v'', u', u''} V W_1
inst_6 : F_1.LaxMonoidal
W : Type u''
inst_7 : Category.{v'', u''} W
inst_8 : MonoidalCategory.{v'', u''} W
F : Functor.{v', v'', u', u''} V W
inst_9 : F.LaxMonoidal
D : Type u
inst_10 : EnrichedCategory.{v', u', u} V D
e :
  (v : V) →
    Equiv.{v' + 1, v'' + 1} (tensorUnit.{v', u'} V (𝒞 := inst) ⟶ v) (tensorUnit.{v'', u''} W (𝒞 := inst_7) ⟶ F.obj v)
h :
  ∀ (v : V) (f : tensorUnit.{v', u'} V (𝒞 := inst) ⟶ v),
    Eq.{v'' + 1} (α := tensorUnit.{v'', u''} W (𝒞 := inst_7) ⟶ F.obj v)
      ((e v : (a : 𝟙_ V ⟶ v) → 𝟙_ W ⟶ F.obj v) f : tensorUnit.{v'', u''} W (𝒞 := inst_7) ⟶ F.obj v)
      (Functor.LaxMonoidal.ε F ≫ F.map f : tensorUnit.{v'', u''} W (𝒞 := inst_7) ⟶ F.obj v)
X Y Z : ForgetEnrichment.{v'', u'', u} W (TransportEnrichment F D)
f : X ⟶ Y
g : Y ⟶ Z
e_f :
  Iso.inv (λ_ (tensorUnit.{v'', u''} W (𝒞 := inst_7))) ≫
      (ForgetEnrichment.homTo W f ⊗ₘ ForgetEnrichment.homTo W g) ≫
        Functor.LaxMonoidal.μ F (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y))
            (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z)) ≫
          F.map (eComp V (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z)) =
    Functor.LaxMonoidal.ε F ≫
      CategoryStruct.comp (Y :=
        F.obj (tensorObj.{v', u'} (𝒞 := inst) (tensorUnit.{v', u'} V (𝒞 := inst)) (tensorUnit.{v', u'} V (𝒞 := inst))))
        (F.map (Iso.inv (λ_ (tensorUnit.{v', u'} V (𝒞 := inst)))))
        (CategoryStruct.comp (Y :=
          F.obj
            (tensorObj.{v', u'} (𝒞 := inst)
              (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y))
              (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z))))
          (F.map
            ((Equiv.symm (e (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y))) :
                  (a : 𝟙_ W ⟶ F.obj (ForgetEnrichment.to W X ⟶[V] ForgetEnrichment.to W Y)) →
                    𝟙_ V ⟶ ForgetEnrichment.to W X ⟶[V] ForgetEnrichment.to W Y)
                (ForgetEnrichment.homTo W f) ⊗ₘ
              (Equiv.symm (e (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z))) :
                  (a : 𝟙_ W ⟶ F.obj (ForgetEnrichment.to W Y ⟶[V] ForgetEnrichment.to W Z)) →
                    𝟙_ V ⟶ ForgetEnrichment.to W Y ⟶[V] ForgetEnrichment.to W Z)
                (ForgetEnrichment.homTo W g)))
          (F.map (eComp V (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z))))
⊢ ForgetEnrichment.homTo W (f ≫ g) =
    Functor.LaxMonoidal.ε F ≫
      F.map
        (CategoryStruct.comp (Y :=
          tensorObj.{v', u'} (𝒞 := inst) (tensorUnit.{v', u'} V (𝒞 := inst)) (tensorUnit.{v', u'} V (𝒞 := inst)))
          (Iso.inv (λ_ (tensorUnit.{v', u'} V (𝒞 := inst))))
          (CategoryStruct.comp (Y :=
            tensorObj.{v', u'} (𝒞 := inst)
              (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y))
              (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z)))
            ((Equiv.symm (e (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y))) :
                  (a : 𝟙_ W ⟶ F.obj (ForgetEnrichment.to W X ⟶[V] ForgetEnrichment.to W Y)) →
                    𝟙_ V ⟶ ForgetEnrichment.to W X ⟶[V] ForgetEnrichment.to W Y)
                (ForgetEnrichment.homTo W f) ⊗ₘ
              (Equiv.symm (e (@Hom.{v', u', u} _ inst inst_1 _ _ (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z))) :
                  (a : 𝟙_ W ⟶ F.obj (ForgetEnrichment.to W Y ⟶[V] ForgetEnrichment.to W Z)) →
                    𝟙_ V ⟶ ForgetEnrichment.to W Y ⟶[V] ForgetEnrichment.to W Z)
                (ForgetEnrichment.homTo W g))
            (eComp V (ForgetEnrichment.to W X) (ForgetEnrichment.to W Y) (ForgetEnrichment.to W Z))))

Tactic:
simp only [ForgetEnrichment.homTo_comp, eComp_eq, Category.assoc, Functor.map_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.ForgetEnrichment.homTo_comp, CategoryTheory.TransportEnrichment.eComp_eq, CategoryTheory.Functor.map_comp