Theorem PowerObject_nt_is_nat_z_iso : is_nat_z_iso PowerObject_nattrans. Proof. intros (a,b). cbn. use make_is_z_isomorphism. + intro g. exact ((identity b) ⨱ g · (PowerObject_inPred b)). + cbn. use make_is_inverse_in_precat. - use funextfun. intros f. cbn. use pathsinv0. use PowerObject_transpose_tri. - use funextfun. intros g. use pathsinv0. use path_to_ctr. apply idpath. Defined.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core.