proposition: Lemma BinCoproductsHSET : BinCoproducts HSET.
proof: Proof. intros A B. use make_BinCoproduct. - apply (setcoprod A B). - simpl in *; apply ii1. - simpl in *; intros x; apply (ii2 x). - apply (make_isBinCoproduct _ HSET). intros C f g; simpl in *. use tpair. * exists (sumofmaps f g); abstract (split; apply idpath). * abstract (intros h; apply subtypePath; [ intros x; apply isapropdirprod; apply has_homsets_HSET | destruct h as [t [ht1 ht2]]; simpl; apply funextfun; intro x; rewrite <- ht2, <- ht1; unfold compose; simpl; case x; intros; apply idpath]). Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Sets UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Coproducts UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Pushouts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Categories.HSET.Core UniMath.Combinatorics.FiniteSets UniMath.MoreFoundations.Subtypes