Definition is_profunctor_nat_iso_weq_is_z_iso {C₁ C₂ : category} {P Q : profunctor C₁ C₂} (τ : profunctor_nat_trans P Q) : is_z_isomorphism (C := [category_binproduct C₂^op C₁ , HSET_univalent_category]) τ ≃ is_profunctor_nat_iso τ. Proof. use weqimplimpl. - exact is_z_iso_to_is_profunctor_nat_iso. - exact is_profunctor_nat_iso_to_is_z_iso. - apply isaprop_is_z_isomorphism. - apply isaprop_is_profunctor_nat_iso. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.HSET.SetCoends. Require Import UniMath.CategoryTheory.Profunctors.Core. Require Import UniMath.CategoryTheory.Profunctors.Examples.