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. 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.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.ProductCategory.