(** The identity algebra homomorphism. *)
  Program Definition Algebra_Hom_id (alg : Algebra) : Algebra_Hom alg alg :=
    {|
      Alg_map := id
    |}.

(* Specifications: *)
Theorem Algebra_Hom_id_unit_left {alg alg' : Algebra} (f : Algebra_Hom alg alg') : (Algebra_Hom_compose f (Algebra_Hom_id alg')) = f.
Theorem Algebra_Hom_id_unit_right {alg alg' : Algebra} (f : Algebra_Hom alg alg') : (Algebra_Hom_compose (Algebra_Hom_id alg) f) = f.