Proposition is_sym_monoidal_adjunction_from_strong : is_sym_monoidal_adjunction V₁ V₂ (monoidal_adjunction_from_strong HL). Proof. split. - exact HHL. - intros x y ; cbn. rewrite !assoc. etrans. { apply maponpaths_2. apply (nat_trans_ax η). } rewrite !assoc'. apply maponpaths. cbn. rewrite <- !functor_comp. apply maponpaths. refine (!_). rewrite !assoc'. etrans. { apply maponpaths. apply tensor_sym_mon_braiding. } rewrite !assoc. apply maponpaths_2. use z_iso_inv_on_right. rewrite !assoc. use z_iso_inv_on_left. cbn. refine (!_). apply HHL. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Monads.Comonads. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Monoidal.FunctorCategories. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations. Section MonoidalAdjunctionFromStrong. End MonoidalAdjunctionFromStrong. Section SymMonoidalAdjunctionFromStrong. End SymMonoidalAdjunctionFromStrong.