Lemma monoidal_swapped_laws {V : category} (Mon_V : monoidal V) : monoidal_laws (monoidal_swapped_data Mon_V). Proof. repeat split. - intro ; intro ; apply (bifunctor_rightid Mon_V). - intro ; intro ; apply (bifunctor_leftid Mon_V). - intro ; intros ; apply (bifunctor_rightcomp Mon_V). - intro ; intros ; apply (bifunctor_leftcomp Mon_V). - intro ; intros ; apply (! bifunctor_equalwhiskers Mon_V _ _ _ _ _ _). - intro ; intros ; apply monoidal_rightunitornat. - apply monoidal_rightunitorisolaw. - apply monoidal_rightunitorisolaw. - intro ; intros ; apply monoidal_leftunitornat. - apply monoidal_leftunitorisolaw. - apply monoidal_leftunitorisolaw. - intro ; intros ; apply (! monoidal_associatorinvnatright Mon_V _ _ _ _ _). - intro ; intros ; apply (! monoidal_associatorinvnatleft Mon_V _ _ _ _ _). - intro ; intros ; apply (! monoidal_associatorinvnatleftright Mon_V _ _ _ _ _). - apply monoidal_associatorisolaw. - apply monoidal_associatorisolaw. - intro ; intros. cbn. rewrite (! monoidal_triangleidentity Mon_V _ _). rewrite assoc. rewrite (pr2 (monoidal_associatorisolaw Mon_V _ _ _)). apply id_left. - intro ; intros ; apply monoidal_pentagon_identity_inv. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.PartA. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.opp_precat. Import BifunctorNotations. Import MonoidalNotations. Section EquivalenceFromTensorWithUnit. Import MonoidalNotations. End EquivalenceFromTensorWithUnit. Import MonoidalNotations. Import MonoidalNotations. Import MonoidalNotations. Import MonoidalNotations. Import MonoidalNotations.