proposition: Lemma catiso_to_precategory_data_path {A B : precategory} (F : catiso A B) : precategory_data_from_precategory A = precategory_data_from_precategory B.
proof: Proof. destruct A as [[[Ao Am] [Ai Ac]] Aax]. destruct B as [[[Bo Bm] [Bi Bc]] Bax]. eapply total2_paths_b. Unshelve. 2: { simpl. exact (catiso_to_precategory_ob_mor_path F). } apply pathsinv0. eapply pathscomp0. apply (transportb_dirprod _ _ _ _ _ (catiso_to_precategory_ob_mor_path F)). apply dirprodeq. - apply funextsec. intros a. apply (catiso_to_precategory_id_path F). - apply funextsec. intro. apply funextsec. intro. apply funextsec. intro. apply (catiso_to_precategory_comp_path F). 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