Definition is_profunctor_nat_iso_to_is_z_iso {C₁ C₂ : category} {P Q : profunctor C₁ C₂} {τ : profunctor_nat_trans P Q} (Hτ : is_profunctor_nat_iso τ) : is_z_isomorphism (C := [category_binproduct C₂^op C₁ , HSET_univalent_category]) τ. Proof. use make_is_z_isomorphism. - exact (inv_profunctor_nat_trans Hτ). - split. + abstract (use eq_profunctor_nat_trans ; intros y x h ; cbn ; apply inv_profunctor_nat_iso_left). + abstract (use eq_profunctor_nat_trans ; intros y x h ; cbn ; apply inv_profunctor_nat_iso_right). 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.