Definition PowerObject_from_exponentials : PowerObject Prod Ω. Proof. use make_PowerObject. + intro b. exact (ExpFun b Ω). + intro b. use (ExpEv b). + intros c b f. use make_iscontr. - split with (φ_adj (ExpAdj b) f). use pathsinv0. use (pathscomp0 (b:=(φ_adj_inv (ExpAdj b) (φ_adj (ExpAdj b) f)))). * apply idpath. * use φ_adj_inv_after_φ_adj. - intros (t,tis). use subtypePath. * intro. use homset_property. * use (invmaponpathsweq (invweq (adjunction_hom_weq (ExpAdj b) c Ω))). cbn. rewrite φ_adj_inv_after_φ_adj. use pathsinv0. use tis. 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.