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

(* Specifications: *)
(** Identity algebra homomorphism is the left unit of compositon. *)
Theorem Algebra_Hom_id_unit_left
{alg alg' : Algebra}
(f : Algebra_Hom alg alg') :
(Algebra_Hom_compose f (Algebra_Hom_id alg')) = f.
(** Identity algebra homomorphism is the right unit of compositon. *)
Theorem Algebra_Hom_id_unit_right
{alg alg' : Algebra}
(f : Algebra_Hom alg alg') :
(Algebra_Hom_compose (Algebra_Hom_id alg) f) = f.