Lemma Id_in_sub_to_iso_equal_iso (a b : ob (full_sub_category C C')) : Id_in_sub_to_iso a b = funcomp (total2_paths_hProp_equiv C' a b) (@idtoiso _ (pr1 a) (pr1 b)). Proof. apply funextfun. intro p. destruct p. apply z_iso_eq. simpl; apply 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.