Proposition enriched_relation_comp_fun_left {X Y Z : hSet} (f : X → Y) (R : enriched_relation V Y Z) : f^* ·e R = fun_comp_enriched_relation R f (idfun _). Proof. use eq_enriched_relation. - intros x z. use CoproductArrow. intro y. refine (sym_mon_braiding _ _ _ · _). use arrow_from_tensor_coproduct_benabou_cosmos. intro p ; cbn. induction p. apply mon_runitor. - intros x z. exact (mon_linvunitor _ · (CoproductIn _ _ _ (idpath _) #⊗ identity _) · CoproductIn _ _ _ (f x)). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.PosetCat. Require Import UniMath.CategoryTheory.Limits.Coends. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.