Proposition lmap_e_arr_lmap_e {V : sym_mon_closed_cat} {C₁ C₂ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} (P : E₁ ↛e E₂) (x : C₁) {y₁ y₂ : C₂} (g : y₂ --> y₁) (y₃ : C₂) : (identity _ #⊗ lmap_e_arr P x g) · lmap_e P y₂ y₃ x = (postcomp_arr E₂ y₃ g #⊗ identity _) · lmap_e P y₁ y₃ x. Proof. unfold lmap_e_arr. rewrite tensor_comp_id_l. rewrite !assoc'. rewrite lmap_e_lmap_e. rewrite !assoc. apply maponpaths_2. unfold postcomp_arr. rewrite !tensor_comp_id_r. apply maponpaths_2. rewrite tensor_comp_id_l. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. rewrite <- tensor_comp_id_r. rewrite tensor_sym_mon_braiding. rewrite tensor_comp_id_r. apply idpath. } rewrite !assoc. apply maponpaths_2. rewrite mon_inv_triangle. etrans. { apply maponpaths_2. rewrite !assoc'. rewrite mon_lassociator_rassociator. apply id_right. } rewrite <- tensor_comp_id_r. rewrite sym_mon_braiding_rinvunitor. 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.