proposition: Lemma isBinProduct_to_isBinProduct' : isBinProduct _ _ _ p p1 p2 -> isBinProduct'.
proof: Proof. intros isBP ? fg. unfold hfiber, postcomp_with_projections. apply (@iscontrweqf (∑ u : C ⟦ a, p ⟧, u · p1 = pr1 fg × u · p2 = pr2 fg)). - use weqfibtototal; intro; cbn. apply invweq, pathsdirprodweq. - exact (isBP a (pr1 fg) (pr2 fg)). 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