Lemma U_reflects_isos (n m : Δ) ( f : n --> m ) : (is_z_isomorphism ((# U) f)) -> (is_z_isomorphism f). Proof. intro I. use tpair. { exact (is_z_isomorphism_mor I,, mon_iso_mon_inv I). } cbv beta. exact (fincard_inv_is_inverse_in_precat n m f I). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.FiniteSets. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.CategoryTheory.SimplicialSets.