Proposition profunctor_nat_trans_lmap {C₁ C₂ : category} {P Q : profunctor C₁ C₂} (τ : profunctor_nat_trans P Q) {y₁ y₂ : C₂} {x : C₁} (g : y₂ --> y₁) (h : P y₁ x) : τ y₂ x (lmap P g h) = lmap Q g (τ y₁ x h). Proof. rewrite lmap_functor. rewrite (profunctor_nat_trans_natural τ g (identity _) h). apply idpath. 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.