Proposition comp_profunctor_mor_comm {z₁ z₂ : C₃} (k : z₂ --> z₁) {x₁ x₂ : C₁} (f : x₁ --> x₂) {y : C₂} (h : P y x₁) (h' : Q z₁ y) : comp_profunctor_mor k f (comp_profunctor_ob_in _ h h') = comp_profunctor_ob_in y (rmap P f h) (lmap Q k h'). Proof. unfold comp_profunctor_mor. rewrite from_comp_profunctor_ob_comm. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.HSET.SetCoends. Require Import UniMath.CategoryTheory.Profunctors.Core.