Proposition enriched_profunctor_coend_comm (x y : C) : lmap_e P y x x · enriched_profunctor_coend_in x = rmap_e P y x y · enriched_profunctor_coend_in y. Proof. pose (maponpaths (λ z, CoproductIn _ _ mor_coprod (x ,, y) · z) (CoequalizerEqAr enriched_profunctor_coend_coequalizer)) as p. cbn in p. unfold left_mor, right_mor in p. rewrite !assoc in p. refine (_ @ p @ _). - refine (!_). rewrite (CoproductInCommutes _ _ _ mor_coprod _ _ (x ,, y)). rewrite !assoc'. apply idpath. - rewrite (CoproductInCommutes _ _ _ mor_coprod _ _ (x ,, y)). rewrite !assoc'. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Profunctor. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.