Proposition inv_profunctor_nat_trans_left {C₁ C₂ : category} {P Q : profunctor C₁ C₂} {τ : profunctor_nat_trans P Q} (Hτ : is_profunctor_nat_iso τ) : profunctor_nat_trans_comp (inv_profunctor_nat_trans Hτ) τ = profunctor_nat_trans_id _. Proof. use eq_profunctor_nat_trans. intros y x h ; cbn. apply (homotweqinvweq (make_weq _ (Hτ y x))). Qed.
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.