Proposition rmap_e_arr_rmap_e {V : sym_mon_closed_cat} {C₁ C₂ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} (P : E₁ ↛e E₂) {x₁ x₂ : C₁} (f : x₁ --> x₂) (x₃ : C₁) (y : C₂) : (identity _ #⊗ rmap_e_arr P f y) · rmap_e P y x₂ x₃ = (precomp_arr E₁ x₃ f #⊗ identity _) · rmap_e P y x₁ x₃. Proof. unfold rmap_e_arr. rewrite tensor_comp_id_l. rewrite !assoc'. rewrite rmap_e_rmap_e. rewrite !assoc. apply maponpaths_2. unfold precomp_arr. rewrite !tensor_comp_id_r. apply maponpaths_2. rewrite tensor_comp_id_l. rewrite !assoc'. rewrite tensor_rassociator. rewrite !assoc. apply maponpaths_2. rewrite mon_inv_triangle. rewrite !assoc'. rewrite mon_lassociator_rassociator. apply id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Enriched.Enriched. Require Import UniMath.CategoryTheory.EnrichedCats.Enriched.EnrichmentEquiv. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.Tensor. Require Import UniMath.CategoryTheory.EnrichedCats.Bifunctors.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.SelfEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.OppositeEnriched. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations. Section FromEnrichedProfunctor. End FromEnrichedProfunctor.