Proposition rassociator_enriched_profunctor_data_eq (z : C₄) (y₁ y₂ : C₃) (w : C₁) : (identity _ #⊗ comp_enriched_profunctor_lmap P₁ P₂ y₂ y₁ w · sym_mon_braiding _ _ _ · internal_eval _ _) #⊗ identity _ · (rassociator_enriched_profunctor_mor z y₁ w #⊗ identity _ · internal_eval _ _) = sym_mon_braiding _ _ _ #⊗ identity _ · mon_lassociator _ _ _ · identity _ #⊗ rmap_e P₃ z y₁ y₂ · (rassociator_enriched_profunctor_mor z y₂ w #⊗ identity _ · internal_eval _ _). Proof. use is_inj_internal_lam. use is_inj_internal_transpose. unfold internal_transpose. use from_comp_enriched_profunctor_eq. intros x'. use internal_funext. intros a h. rewrite !tensor_comp_r_id_r. rewrite !(tensor_split (comp_enriched_profunctor_in _ _ _ _ _) h). rewrite !assoc'. apply maponpaths. rewrite !internal_beta. rewrite !assoc. rewrite !tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. clear a h. use internal_funext. intros a h. rewrite !tensor_comp_r_id_r. rewrite !(tensor_split (identity _ #⊗ comp_enriched_profunctor_in _ _ _ _ _) h). rewrite !assoc'. apply maponpaths. clear a h. rewrite !internal_beta. rewrite !assoc. rewrite <- !tensor_comp_id_r. etrans. { do 3 apply maponpaths_2. 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. etrans. { rewrite !assoc'. do 3 rewrite tensor_comp_id_r. rewrite !assoc'. rewrite rassociator_enriched_profunctor_mor_comm. unfold rassociator_enriched_profunctor_mor_ob. rewrite !assoc. apply idpath. } refine (!_). etrans. { rewrite tensor_sym_mon_braiding. rewrite tensor_comp_id_r. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_id_id. rewrite <- tensor_split'. rewrite tensor_split. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite rassociator_enriched_profunctor_mor_comm. apply idpath. } unfold rassociator_enriched_profunctor_mor_ob. rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. do 2 apply maponpaths. rewrite <- tensor_id_id. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. rewrite <- tensor_comp_id_l. rewrite comp_enriched_profunctor_comm'. rewrite !tensor_comp_id_l. apply idpath. } rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { rewrite !assoc'. do 2 apply maponpaths. rewrite tensor_comp_id_l. rewrite tensor_comp_id_r. rewrite !assoc'. rewrite tensor_lassociator. apply idpath. } rewrite !assoc. apply maponpaths_2. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite mon_lassociator_lassociator. rewrite !assoc'. apply maponpaths. rewrite tensor_lassociator. apply maponpaths. rewrite <- !tensor_comp_id_l. apply maponpaths. refine (!_). etrans. { do 2 apply maponpaths. rewrite !assoc. apply sym_mon_hexagon_rassociator. } etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. rewrite id_left. apply idpath. } rewrite !assoc. rewrite mon_lassociator_rassociator. apply id_left. 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.