Proposition internal_assoc (w x y z : V) : (internal_comp x y z #⊗ identity _) · internal_comp w x z = mon_lassociator _ _ _ · (identity _ #⊗ internal_comp w x y) · internal_comp w y z. Proof. use internal_funext. intros a h. etrans. { rewrite tensor_comp_r_id_r. rewrite !assoc'. apply maponpaths. apply internal_beta. } etrans. { rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_mor. rewrite id_right. rewrite tensor_split. rewrite !assoc'. apply maponpaths. apply internal_beta. } refine (!_). etrans. { rewrite tensor_comp_r_id_r. rewrite !assoc'. apply maponpaths. apply internal_beta. } rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. rewrite tensor_comp_r_id_r. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. rewrite <- tensor_comp_id_l. do 2 apply maponpaths. apply internal_beta. } rewrite !tensor_comp_id_l. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. refine (!_). etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite <- tensor_id_id. rewrite tensor_lassociator. rewrite !tensor_comp_id_l. apply idpath. } rewrite !assoc. apply maponpaths_2. rewrite mon_lassociator_lassociator. rewrite !assoc'. rewrite <- tensor_comp_id_l. rewrite <- tensor_lassociator. rewrite tensor_comp_id_l. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite <- tensor_lassociator. rewrite !assoc. apply maponpaths_2. rewrite <- tensor_comp_mor. rewrite id_left. rewrite !tensor_id_id. rewrite id_right. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import MonoidalNotations.