End join_hom_fun.
Arguments lookup [Key A] _ _ _.

Lemma join_hom_identity  {A}{B}`{Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A}`{Perm_alg B}{SB: Sep_alg B}{CB: Disj_alg B}:
forall (f: A -> B) a1, identity a1 -> join_hom f -> identity (f a1).