Lemma iso_in_precat_is_iso_in_subcat (a b : ob (full_sub_category C C')) (f : z_iso (pr1 a) (pr1 b)) : is_z_isomorphism (C:=full_sub_category C C') (precategory_morphisms_in_subcat f tt). Proof. exists (precategory_morphisms_in_subcat (inv_from_z_iso f) tt). split; simpl. - apply eq_in_sub_precategory; simpl. apply z_iso_inv_after_z_iso. - apply eq_in_sub_precategory; simpl. apply z_iso_after_z_iso_inv. 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.