∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] {D : Type u_6} {E : Type u_4}
  [inst_1 : CategoryTheory.Category.{u_5, u_6} D] [inst_2 : CategoryTheory.Category.{u_3, u_4} E]
  (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) {X Y : CategoryTheory.WithInitial C} (f : X ⟶ Y),
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.WithInitial.map (F.comp G)).map f)
      ((fun X =>
            match X with
            | CategoryTheory.WithInitial.of a =>
              CategoryTheory.Iso.refl
                ((CategoryTheory.WithInitial.map (F.comp G)).obj (CategoryTheory.WithInitial.of a))
            | CategoryTheory.WithInitial.star =>
              CategoryTheory.Iso.refl ((CategoryTheory.WithInitial.map (F.comp G)).obj CategoryTheory.WithInitial.star))
          Y).hom =
    CategoryTheory.CategoryStruct.comp
      ((fun X =>
            match X with
            | CategoryTheory.WithInitial.of a =>
              CategoryTheory.Iso.refl
                ((CategoryTheory.WithInitial.map (F.comp G)).obj (CategoryTheory.WithInitial.of a))
            | CategoryTheory.WithInitial.star =>
              CategoryTheory.Iso.refl ((CategoryTheory.WithInitial.map (F.comp G)).obj CategoryTheory.WithInitial.star))
          X).hom
      (((CategoryTheory.WithInitial.map F).comp (CategoryTheory.WithInitial.map G)).map f)