Definition U_action_ρ_is_nat_z_iso : is_nat_z_iso U_action_ρ_nat_trans. Proof. intro. cbn. use is_z_isomorphism_comp. - use is_z_iso_tensor_z_iso. + exact (identity_is_z_iso _ ). + apply (is_z_iso_inv_from_z_iso (make_z_iso _ _ (strong_monoidal_functor_ϵ_is_z_iso U))). - exact (pr2 ρ_A c). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.Bicategories.MonoidalCategories.EndofunctorsMonoidal.