Proposition rmap_e_arr_comp {V : sym_mon_closed_cat} {C₁ C₂ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} (P : E₁ ↛e E₂) {x₁ x₂ x₃ : C₁} (f₁ : x₁ --> x₂) (f₂ : x₂ --> x₃) (y : C₂) : rmap_e_arr P (f₁ · f₂) y = rmap_e_arr P f₁ y · rmap_e_arr P f₂ y. Proof. unfold rmap_e_arr. rewrite enriched_from_arr_comp. rewrite !assoc. rewrite tensor_comp_id_r. rewrite !assoc'. rewrite rmap_e_comp. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. rewrite tensor_comp_id_r. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. rewrite <- tensor_comp_mor. rewrite id_right. apply idpath. } refine (!_). etrans. { apply maponpaths. rewrite !assoc. rewrite tensor_linvunitor. rewrite !assoc'. apply maponpaths. rewrite <- tensor_comp_mor. rewrite id_left. rewrite id_right. apply idpath. } rewrite !assoc. apply maponpaths_2. rewrite !assoc'. apply maponpaths. rewrite <- mon_linvunitor_triangle. apply idpath. 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.