Definition sym_mon_closed_left_tensor_right_adjoint_universal (V : sym_mon_closed_cat) (x y : V) : is_universal_arrow_from (monoidal_right_tensor x) y (x ⊸ y) (internal_eval x y). Proof. intros z f. use iscontraprop1. - abstract (use invproofirrelevance ; intros φ₁ φ₂ ; use subtypePath ; [ intro ; apply homset_property | ] ; refine (internal_eta _ @ _ @ !(internal_eta _)) ; apply maponpaths ; exact (!(pr2 φ₁) @ pr2 φ₂)). - refine (internal_lam f ,, _). abstract (cbn ; rewrite internal_beta ; apply idpath). Defined.
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.