Section Exp_Cat_morph_ex_compose.
Context {C C' C'' : Category}
(F : (C'' × C) –≻  C')
{B : Category}
(G : B –≻ C'')
.

(** This is the more specific case of curry_compose. Proven separately for cat because of universe polymorphism issues that prevent cat to both have expoenentials and type_cat in it. *)
Theorem Exp_Cat_morph_ex_compose :
Exp_Cat_morph_ex (F ∘ (Prod_Functor G (Functor_id C)))
= (Exp_Cat_morph_ex F) ∘ G.