Lemma is_mon_nat_trans_pointwise_inverse (isnziα : is_nat_z_iso α) : is_mon_nat_trans Fm Gm α -> is_mon_nat_trans Gm Fm (nat_z_iso_inv (α,,isnziα)). Proof. intro ismnt. split. - intros x y. cbn. unfold fmonoidal_preservestensordata. set (aux := (_,, is_z_iso_bifunctor_z_iso N _ _ (isnziα x) (isnziα y)) : z_iso _ _). apply pathsinv0, (z_iso_inv_on_right _ _ _ aux). rewrite assoc. apply (z_iso_inv_on_left _ _ _ _ (_,,isnziα (x ⊗_{ M} y))). cbn. apply (!(pr1 ismnt x y)). - cbn. apply pathsinv0, (z_iso_inv_on_left _ _ _ _ (_,,isnziα I_{M})). apply (!(pr2 ismnt)). 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.whiskering. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations.