Proposition internal_beta {x y z : V} (f : z ⊗ x --> y) : internal_lam f #⊗ identity x · internal_eval x y = f. Proof. unfold internal_lam. rewrite tensor_comp_r_id_l. rewrite !assoc'. unfold internal_eval. etrans. { apply maponpaths. pose (nat_trans_ax (counit_from_are_adjoints (pr2 (pr2 V x))) _ _ f) as p. refine (_ @ p). apply maponpaths_2. cbn. unfold rightwhiskering_on_morphisms. unfold monoidal_cat_tensor_mor. unfold functoronmorphisms1. refine (_ @ id_right _). apply maponpaths. apply (bifunctor_leftid V). } cbn. rewrite !assoc. refine (_ @ id_left _). apply maponpaths_2. refine (_ @ triangle_id_left_ad (pr2 (pr2 V x)) _). apply maponpaths_2. cbn. unfold rightwhiskering_on_morphisms. unfold monoidal_cat_tensor_mor. unfold functoronmorphisms1. refine (_ @ id_right _). apply maponpaths. apply (bifunctor_leftid V). 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.