Definition cmonoidal_unit_noncurried_hom_data (M N : ob UMONCAT) : nat_trans_data (functor_identity (hom M N)) (cmonoidal_to_noncurried_hom M N ∙ cmonoidal_from_noncurried_hom M N). Proof. intro F. exists (nat_trans_id _). use tpair. - use tpair. + use tpair. * intros x y. rewrite id_right. refine (! id_left _ @ _). apply maponpaths_2. etrans. 2: { apply (! tensor_on_hom_eq N _ _). } exact (! functor_id (pr121 (cmonoidal_to_noncurriedmonoidal N)) (pr11 F x , pr11 F y)). * apply id_right. + repeat split. - exact tt. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorCategory. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.EquivalenceWhiskeredNonCurriedMonoidalCategories. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesReordered. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.AssociatorUnitorsLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.FinalLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.AssociatorUnitorsLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.FinalLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.EquivalenceMonCatCurried. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.EquivalenceWhiskeredCurried.