proposition: Lemma correct_hom_on_comp {A B : precategory} (F : catiso A B) : forall a a' a'', forall f : F a --> F a', forall g : F a' --> F a'', (eqweqmap (correct_hom F _ _)) f · (eqweqmap (correct_hom F _ _)) g = (eqweqmap (correct_hom F _ _)) (f · g).
proof: Proof. intros a a' a'' f g. rewrite !eqweq_correct_hom_is_comp. induction (eqweq_ob_path_is_functor_app F a). induction (eqweq_ob_path_is_functor_app F a'). induction (eqweq_ob_path_is_functor_app F a''). rewrite 3 id_left. simpl. rewrite 3 id_right. reflexivity. Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.UnivalenceAxiom UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful