Proposition enriched_relation_comp_conj {W X Y Z : hSet} (R : enriched_relation V X Y) (f : W → X) (g : Z → Y) : f^* ·e (R ·e g^o) = fun_comp_enriched_relation R f g. Proof. use eq_enriched_relation. - intros w z. use CoproductArrow. intro x. use arrow_from_tensor_coproduct_benabou_cosmos. intro y. refine (sym_mon_braiding _ _ _ · _). use arrow_from_tensor_coproduct_benabou_cosmos. intro p ; cbn. induction p. refine (mon_runitor _ · _). use arrow_from_tensor_coproduct_benabou_cosmos. intro q ; cbn. induction q. apply mon_runitor. - intros w z. refine (_ · CoproductIn _ _ _ (f w)) ; cbn. refine (_ · (CoproductIn _ _ _ (idpath _) #⊗ identity _)). refine (mon_linvunitor _ · (identity _ #⊗ _)). refine (_ · CoproductIn _ _ _ (g z)). refine (mon_rinvunitor _ · (identity _ #⊗ _)). exact (CoproductIn _ _ _ (idpath _)). 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.