Proposition identity_enriched_profunctor_laws : enriched_profunctor_laws identity_enriched_profunctor_data. Proof. repeat split. - intros x y ; cbn. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. rewrite <- enrichment_id_right. apply sym_mon_braiding_runitor. - intros x y ; cbn. rewrite <- enrichment_id_left. apply idpath. - intros x₁ x₂ x₃ y ; cbn. etrans. { rewrite tensor_comp_id_r. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. rewrite tensor_sym_mon_braiding. apply idpath. } rewrite !assoc'. rewrite enrichment_assoc'. apply idpath. } rewrite !assoc. apply maponpaths_2. rewrite tensor_comp_id_l. rewrite !assoc'. rewrite tensor_sym_mon_braiding. rewrite !assoc. apply maponpaths_2. rewrite sym_mon_tensor_lassociator. rewrite !assoc. apply maponpaths_2. rewrite tensor_sym_mon_braiding. apply maponpaths_2. apply sym_mon_tensor_rassociator. - intros x y₁ y₂ y₃ ; cbn. rewrite enrichment_assoc. apply idpath. - intros x₁ x₂ y₁ y₂ ; cbn. rewrite tensor_comp_id_l. rewrite !assoc'. rewrite enrichment_assoc'. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite tensor_sym_mon_braiding. rewrite !assoc. apply maponpaths_2. rewrite sym_mon_tensor_lassociator. rewrite !assoc. do 2 apply maponpaths_2. rewrite mon_lassociator_rassociator. rewrite id_left. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. 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.