proposition: Lemma bincoprod_antidistributor_nat (cc'1 cc'2 : category_binproduct C C) (g : category_binproduct C C ⟦ cc'1, cc'2 ⟧) : bincoprod_antidistributor (pr1 cc'1) (pr2 cc'1) · #F (#(bincoproduct_functor BCPC) g) = #(bincoproduct_functor BCPD) (#(pair_functor F F) g) · bincoprod_antidistributor (pr1 cc'2) (pr2 cc'2).
proof: Proof. etrans. { apply postcompWithBinCoproductArrow. } etrans. 2: { apply pathsinv0, precompWithBinCoproductArrow. } apply maponpaths_12. - etrans. { apply pathsinv0, functor_comp. } etrans. 2: { apply functor_comp. } apply maponpaths. apply BinCoproductIn1Commutes. - etrans. { apply pathsinv0, functor_comp. } etrans. 2: { apply functor_comp. } apply maponpaths. apply BinCoproductIn2Commutes. Qed.
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