Lemma leftclosed_symmetric_is_rightclosed {C : category} (M : monoidal C) : symmetric M -> monoidal_leftclosed M -> monoidal_rightclosed M. Proof. intros B LC. intro x. exists (monoidal_leftclosed_exp LC x). apply (adj_closed_under_nat_z_iso (F2 := rightwhiskering_functor M x)). - apply symmetric_whiskers_swap_nat_z_iso. exact B. - exact (pr2 (LC x)). 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.