proposition: Lemma Exponentials_HSET : Exponentials BinProductsHSET.
proof: Proof. intro a. exists (exponential_functor a). use tpair. - use tpair. + use tpair. * intro x; simpl; apply flip, make_dirprod. * abstract (intros x y f; apply idpath). + use tpair. * intros x xf; simpl in *; apply (pr2 xf (pr1 xf)). * abstract (intros x y f; apply idpath). - abstract (use tpair; [ now intro x; simpl; apply funextfun; intro ax; apply idpath | now intro b; apply funextfun; intro f]). Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.MoreFoundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.AxiomOfChoice UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.slicecat UniMath.CategoryTheory.NNO UniMath.CategoryTheory.SubobjectClassifier UniMath.CategoryTheory.Categories.Type.Core UniMath.CategoryTheory.Categories.Type.MonoEpiIso UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.covyoneda UniMath.CategoryTheory.EpiFacts UniMath.CategoryTheory.Monics UniMath.CategoryTheory.SplitMonicsAndEpis UniMath.CategoryTheory.ElementaryTopos UniMath.CategoryTheory.PowerObject UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.Limits UniMath.CategoryTheory.Categories.HSET.MonoEpiIso UniMath.CategoryTheory.Categories.HSET.Univalence