Definition cmonoidal_formequivalence_noncurried (M N : ob UMONCAT) : forms_equivalence (cmonoidal_to_noncurried_hom M N ,, cmonoidal_from_noncurried_hom M N,, cmonoidal_unit_noncurried_hom M N,, cmonoidal_counit_noncurried_hom M N). Proof. split ; intro. - use tpair. + repeat (use tpair). * apply nat_trans_id. * apply nat_trans_id. * intros x y. simpl. rewrite id_right. etrans. 2: { apply maponpaths_2. apply (! tensor_id _ _ _). } apply (! id_left _). * apply id_right. * exact tt. * exact tt. * exact tt. * exact tt. + split ; apply UMONCAT_2cell_equality ; intro ; apply id_right. - use tpair. + repeat (use tpair). * apply nat_trans_id. * apply nat_trans_id. * intros x y. simpl. rewrite id_right. etrans. { apply maponpaths_2. apply (! tensor_on_hom_eq N _ _). } etrans. { apply maponpaths_2. apply (tensor_id _ _ _). } apply id_left. * apply id_right. * exact tt. * exact tt. * exact tt. + split. * apply LaxMonoidalFunctor_mor_equality. intro ; apply id_right. * apply LaxMonoidalFunctor_mor_equality. intro ; apply id_right. 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.