Lemma are_adjoints_constprod_functor2 A : are_adjoints (constprod_functor2 BinProductsHSET A) (exponential_functor A). Proof. use tpair. - use tpair. + use tpair. * intro x; simpl; apply make_dirprod. * abstract (intros x y f; apply idpath). + use tpair. * intros X fx; apply (pr1 fx (pr2 fx)). * abstract (intros x y f; apply idpath). - abstract (use tpair; [ intro x; simpl; apply funextfun; intro ax; apply idpath | intro b; apply funextfun; intro f; apply idpath]). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.AxiomOfChoice. 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.Adjunctions.Core. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.CategoryTheory.Arithmetic.NNO. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.Categories.Type.Core. Require Import UniMath.CategoryTheory.Categories.Type.MonoEpiIso. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.covyoneda. Require Import UniMath.CategoryTheory.EpiFacts. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.SplitMonicsAndEpis. Require Import UniMath.CategoryTheory.ElementaryTopos. Require Import UniMath.CategoryTheory.PowerObject. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. assert (p : (BinProductsFromPullbacks PullbacksHSET TerminalHSET) = BinProductsHSET).