proposition: Lemma catiso_to_precategory_mor_path {A B : precategory_data} (F : catiso A B) : forall a a', (precategory_morphisms (C:=A)) a a' = transportb (λ T, T -> T -> UU) (catiso_to_precategory_ob_path F) (precategory_morphisms (C:=B)) a a'.
proof: Proof. intros a. intros a'. eapply pathscomp0. apply (catiso_fully_faithful_path F). eapply pathscomp0. apply correct_hom. eapply pathscomp0. apply transport_mor. 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