Definition is_iso_full_sub {C : category} {P : hsubtype C} {x y : full_sub_category C P} {f : x --> y} (Hf : is_z_isomorphism (pr1 f)) : is_z_isomorphism f. Proof. exists (inv_from_z_iso (_,,Hf) ,, tt). split. + abstract (use subtypePath ; [ intro ; apply isapropunit | ] ; exact (z_iso_inv_after_z_iso (_,,Hf))). + abstract (use subtypePath ; [ intro ; apply isapropunit | ] ; exact (z_iso_after_z_iso_inv (_,,Hf))). Defined.
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.