Proposition enriched_triangle_law {V : benabou_cosmos} {C₁ C₂ C₃ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {E₃ : enrichment C₃ V} (P : E₁ ↛e E₂) (Q : E₂ ↛e E₃) : enriched_profunctor_comp_transformation (lassociator_enriched_profunctor P (identity_enriched_profunctor E₂) Q) (rwhisker_enriched_profunctor (runitor_enriched_profunctor P) Q) = lwhisker_enriched_profunctor P (lunitor_enriched_profunctor Q). Proof. use eq_enriched_profunctor_transformation ; intros z x ; cbn. use from_comp_enriched_profunctor_eq. intro y. etrans. { rewrite !assoc. apply maponpaths_2. apply lassociator_enriched_profunctor_data_comm. } refine (!_). etrans. { apply lwhisker_enriched_profunctor_mor_comm. } use is_inj_internal_transpose. use from_comp_enriched_profunctor_eq. intros y'. use internal_funext. intros v h. rewrite !tensor_comp_r_id_r. rewrite !(tensor_split (comp_enriched_profunctor_in _ _ _ _ _) h). rewrite !assoc'. apply maponpaths. clear v h. unfold internal_transpose. rewrite !internal_beta. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. etrans. { rewrite !assoc. apply maponpaths_2. rewrite <- tensor_comp_id_l. apply maponpaths. apply lunitor_enriched_profunctor_mor_comm. } refine (!_). etrans. { rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. rewrite !assoc. apply maponpaths_2. rewrite <- tensor_comp_id_r. apply lassociator_enriched_profunctor_mor_comm. } unfold lassociator_enriched_profunctor_mor_ob. etrans. { rewrite !assoc. rewrite sym_mon_braiding_inv. rewrite id_left. rewrite !assoc'. do 2 apply maponpaths. apply (rwhisker_enriched_profunctor_mor_comm (runitor_enriched_profunctor P)). } etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. do 2 apply maponpaths_2. apply runitor_enriched_profunctor_mor_comm. } rewrite tensor_comp_id_r. rewrite !assoc'. rewrite comp_enriched_profunctor_comm. rewrite !assoc. apply maponpaths_2. rewrite tensor_comp_id_l. rewrite !assoc. refine (_ @ id_left _). apply maponpaths_2. etrans. { apply maponpaths_2. rewrite !assoc'. do 2 apply maponpaths. rewrite !assoc. apply sym_mon_hexagon_lassociator. } rewrite !assoc'. rewrite <- tensor_comp_id_l. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. rewrite id_right. etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. rewrite id_left. apply idpath. } apply mon_rassociator_lassociator. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. 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.Coend. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.CompositionProf. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.Unitors. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.Associators. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.Whiskering. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.