Theorem PowerObject_nt_is_nat_trans : is_nat_trans HomxO HomP PowerObject_nt_data. Proof. intros (a,b) (a',b') (a'a,b'b). cbn in a'a, b'b. use funextfun. intro f. apply pathsinv0. use path_to_ctr. cbn. rewrite id_right. rewrite <-BinProductOfArrows_idxcomp, !assoc'. rewrite <-(PowerObject_transpose_tri). rewrite !assoc, BinProductOfArrows_comp, id_left, id_right. rewrite (PowerObject_transpose_tri f), assoc, BinProductOfArrows_comp, id_right, <-(PowerObject_transpose_tri f). cbn. apply idpath. Qed.
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.