Definition from_enriched_profunctor_coend (v : V) (fs : ∏ (x : C), P x x --> v) (ps : ∏ (x y : C), lmap_e P y x x · fs x = rmap_e P y x y · fs y) : enriched_profunctor_coend --> v. Proof. use CoequalizerOut. - use CoproductArrow. exact fs. - abstract (use CoproductArrow_eq ; intros xy ; induction xy as [ x y ] ; cbn ; rewrite !assoc ; unfold left_mor, right_mor ; rewrite !(CoproductInCommutes _ _ _ mor_coprod _ _ (x ,, y)) ; cbn ; rewrite !assoc' ; rewrite !(CoproductInCommutes _ _ _ ob_coprod) ; exact (ps x y)). Defined.
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.