Proposition internal_id_left (x y : V) : mon_lunitor _ = internal_id y #⊗ identity _ · internal_comp x y y. Proof. use internal_funext. intros w h. rewrite tensor_comp_r_id_r. rewrite !assoc'. unfold internal_comp. rewrite internal_beta. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. refine (!_). etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_mor. rewrite id_right. rewrite tensor_comp_l_id_l. rewrite !assoc'. apply maponpaths. rewrite tensor_split. rewrite !assoc'. apply maponpaths. unfold internal_id. apply internal_beta. } rewrite tensor_lunitor. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite tensor_lunitor. rewrite !assoc. rewrite mon_lunitor_triangle. rewrite <- tensor_split'. 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.