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

Lemma join_hom2_identity  {A}{B}{C}
`{Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A}
`{Perm_alg B}{SB: Sep_alg B}{CB: Disj_alg B}
`{Perm_alg C}{SC: Sep_alg C}{CC: Disj_alg C}:
forall (g: A -> B -> C) a1 b1,
identity a1 -> identity b1 -> join_hom2 g -> identity (g a1 b1).