Definition sieve_to_sieve_iso : z_iso (slicecat_ob_object _ _ (subtype_to_sieve (sieve_to_subtype f))) (slicecat_ob_object _ _ f). Proof. apply iso_in_subcategory_of_monics_weq. use tpair. - exact sieve_to_sieve_nat_trans. - apply nat_trafo_z_iso_if_pointwise_z_iso. exact sieve_to_sieve_is_iso. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.GrothendieckToposes.Sieves. Require Import UniMath.CategoryTheory.Categories.PreorderCategory.Core. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Subobjects. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.OrderTheory.Preorders. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.slicecat.