Lemma precat_paths_in_sub_as_3_maps (a b : ob (full_sub_category C C')): @idtoiso _ a b = funcomp (Id_in_sub_to_iso a b) (iso_in_sub_from_iso a b). Proof. apply funextfun. intro p; destruct p. apply z_iso_eq; simpl. unfold precategory_morphisms_in_subcat. apply eq_in_sub_precategory, idpath. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Subcategory.Core.