proposition: Lemma correct_hom {A B : precategory_data} (F : catiso A B) : forall a a' : A, F a --> F a' = (eqweqmap (catiso_to_precategory_ob_path F) a) --> (eqweqmap (catiso_to_precategory_ob_path F) a').
proof: Proof. intros a a'. assert (W := (!(homotweqinvweq (univalence _ _)) (catiso_ob_weq F))). exact (maponpaths (λ T, (pr1weq T) a --> (pr1weq T) a') W ). 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