Proposition lassociator_enriched_profunctor_mor_eq (z : C₄) (y₁ y₂ : C₃) (x : C₂) (w : C₁) : lmap_e P₂ y₂ y₁ x #⊗ identity _ · internal_lam (lassociator_enriched_profunctor_mor_ob z y₁ x w) = sym_mon_braiding V _ _ #⊗ identity _ · mon_lassociator _ _ _ · identity _ #⊗ rmap_e P₃ z y₁ y₂ · internal_lam (lassociator_enriched_profunctor_mor_ob z y₂ x w). Proof. use internal_funext. intros a h. rewrite !tensor_comp_r_id_r. rewrite !assoc'. rewrite !internal_beta. rewrite !(tensor_split (_ #⊗ _) h). rewrite !assoc'. apply maponpaths. unfold lassociator_enriched_profunctor_mor_ob. rewrite !assoc. rewrite !tensor_sym_mon_braiding. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. apply idpath. } refine (!_). etrans. { rewrite !assoc. do 3 apply maponpaths_2. rewrite <- !tensor_comp_id_r. rewrite tensor_sym_mon_braiding. apply idpath. } rewrite !assoc'. apply maponpaths. etrans. { rewrite !tensor_comp_id_l. rewrite !assoc'. do 2 apply maponpaths. rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. apply idpath. } etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite tensor_id_id. rewrite <- tensor_split. rewrite tensor_split'. rewrite !assoc'. apply maponpaths. apply (comp_enriched_profunctor_comm' (comp_enriched_profunctor P₁ P₂) P₃). } rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. do 3 apply maponpaths. rewrite !assoc. rewrite <- tensor_split'. rewrite tensor_split. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_id_id. rewrite tensor_rassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. apply maponpaths. rewrite <- tensor_comp_id_r. apply idpath. } rewrite !assoc. etrans. { apply maponpaths_2. etrans. { rewrite !assoc'. do 4 apply maponpaths. rewrite !assoc. apply sym_mon_hexagon_rassociator. } etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. rewrite id_left. apply idpath. } etrans. { do 2 apply maponpaths. rewrite !assoc. rewrite mon_rassociator_rassociator. apply idpath. } apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite mon_lassociator_rassociator. rewrite tensor_id_id. rewrite id_left. apply idpath. } rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. apply maponpaths. rewrite <- !tensor_comp_id_r. apply maponpaths_2. clear z. cbn. etrans. { do 3 apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite tensor_sym_mon_braiding. rewrite !assoc'. rewrite comp_enriched_profunctor_lmap_comm. rewrite !assoc. rewrite sym_mon_braiding_inv. apply id_left. } unfold comp_enriched_profunctor_lmap_mor. rewrite !assoc. apply maponpaths_2. etrans. { do 2 apply maponpaths_2. rewrite !assoc'. rewrite sym_mon_braiding_inv. rewrite id_right. apply idpath. } etrans. { apply maponpaths_2. rewrite !assoc'. rewrite mon_rassociator_lassociator. apply id_right. } rewrite <- tensor_comp_id_l. rewrite !assoc. rewrite sym_mon_braiding_inv. 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.Limits.Coends. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Profunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.StandardProfunctors. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.ProfunctorTransformations. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Invertible. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.ProfunctorSquares. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.CompositionProf. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.