Proposition lax_extension_id_enriched_relation {V : quantale_cosmos} {F : SET ⟶ SET} (L : lax_extension V F) (X : hSet) : id_enriched_relation (F X) ≤e L _ _ (id_enriched_relation X). Proof. rewrite <- !companion_enriched_relation_id. refine (enriched_relation_le_trans _ (lax_extension_enriched_rel_companion L (idfun X))). use enriched_relation_companion_le. refine (!_). exact (functor_id F X). Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichedRelation. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.Limits.Coproducts. Import MonoidalNotations.