∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_4, u_1} C] {D : Type u_3}
  [inst_1 : CategoryTheory.Category.{u_2, u_3} D] (F : CategoryTheory.Functor C D) (X : CategoryTheory.WithInitial C),
  (match X, X, CategoryTheory.CategoryStruct.id X with
    | CategoryTheory.WithInitial.of a, CategoryTheory.WithInitial.of a_1, f => F.map (CategoryTheory.WithInitial.down f)
    | CategoryTheory.WithInitial.star, CategoryTheory.WithInitial.of a, x => PUnit.unit
    | CategoryTheory.WithInitial.star, CategoryTheory.WithInitial.star, x => PUnit.unit) =
    CategoryTheory.CategoryStruct.id
      (match X with
      | CategoryTheory.WithInitial.of x => CategoryTheory.WithInitial.of (F.obj x)
      | CategoryTheory.WithInitial.star => CategoryTheory.WithInitial.star)