Lemma sieve_to_sieve_commutes : slicecat_ob_morphism _ _ (subtype_to_sieve (sieve_to_subtype f)) = sieve_to_sieve_iso · slicecat_ob_morphism _ _ f. Proof. apply Monic_eq. apply nat_trans_eq_alt. intro y. apply funextfun. intro H. apply propproperty. Qed.
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.