Proposition profunctor_nat_trans_natural {C₁ C₂ : category} {P Q : profunctor C₁ C₂} (τ : profunctor_nat_trans P Q) {y₁ y₂ : C₂} {x₁ x₂ : C₁} (g : y₂ --> y₁) (f : x₁ --> x₂) (h : P y₁ x₁) : τ y₂ x₂ (P #[ g , f ] h) = Q #[ g , f ] (τ y₁ x₁ h). Proof. exact (eqtohomot (nat_trans_ax τ (y₁ ,, x₁) (y₂ ,, x₂) (g ,, f)) h). 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.