proposition: Lemma BinCoproducts_from_Colims : Colims_of_shape two_graph C -> BinCoproducts C.
proof: Proof. intros H a b. set (CC := H (bincoproduct_diagram a b)). use make_BinCoproduct. + apply (colim CC). + apply (colimIn CC true). + apply (colimIn CC false). + apply (make_isBinCoproduct _ C); intros c f g. use unique_exists. - apply colimArrow, (BinCoprod f g). - abstract (split; [ apply (colimArrowCommutes CC c (BinCoprod f g) true) | apply (colimArrowCommutes CC c (BinCoprod f g) false) ]). - abstract (intros h; apply isapropdirprod; apply C). - abstract (now intros h [H1 H2]; apply colimArrowUnique; intro x; induction x). Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.ProductCategory