Section Category_comp.
Variable C : category.
Definition comp_hom (a b c : C) (g : Hom b c) (f : Hom a b) :=
Hom_comp a b c (couple g f).

Lemma comp_hom_compatible :
forall (a b c : C) (x x' : Hom b c) (y y' : Hom a b),
Equal x x' -> Equal y y' -> Equal (comp_hom x y) (comp_hom x' y').