Proposition representable_enriched_profunctor_left_laws : enriched_profunctor_laws representable_enriched_profunctor_left_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 !assoc. rewrite <- tensor_comp_id_r. rewrite functor_enrichment_id. 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 !assoc. rewrite <- tensor_comp_id_r. rewrite functor_enrichment_comp. rewrite tensor_comp_r_id_r. rewrite !assoc'. rewrite enrichment_assoc. rewrite !assoc. rewrite tensor_lassociator. apply maponpaths_2. rewrite !assoc'. apply maponpaths. rewrite <- !tensor_comp_mor. rewrite !id_right. rewrite id_left. apply idpath. - intros x₁ x₂ y₁ y₂ ; cbn. rewrite tensor_comp_id_l. rewrite !assoc'. etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite <- tensor_split. rewrite tensor_split'. rewrite !assoc'. rewrite enrichment_assoc'. apply idpath. } rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite tensor_sym_mon_braiding. rewrite tensor_comp_id_r. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite <- tensor_id_id. rewrite tensor_rassociator. 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.