proposition: Lemma iso_to_isBinProduct {x y z : C} (BP : BinProduct C x y) (i : iso z (BinProductObject C BP)) : isBinProduct C _ _ z (i · (BinProductPr1 C BP)) (i · (BinProductPr2 C BP)).
proof: Proof. intros w f g. use unique_exists. - exact ((BinProductArrow C BP f g) · (iso_inv_from_iso i)). - abstract (exact (iso_to_isBinProduct_comm BP i w f g)). - abstract (intro; apply isapropdirprod; apply C). - abstract (intros y0 T; exact (iso_to_isBinProduct_unique BP i w f g y0 T)). 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.Monics UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Products UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.FunctorCategory UniMath.Combinatorics.StandardFiniteSets