Lemma coprod_antidistributor_nat_right (a1 a2 : A) (cs : power_category I C) (f : A ⟦ a1, a2 ⟧) : bifunctor_coprod_antidistributor a1 cs · f ⊗^{F}_{r} coproduct_functor I CPC cs = #(coproduct_functor I CPD) (fun i => f ⊗^{F}_{r} (cs i)) · bifunctor_coprod_antidistributor a2 cs. Proof. etrans. { apply postcompWithCoproductArrow. } etrans. 2: { apply pathsinv0, precompWithCoproductArrow. } apply maponpaths; apply funextsec; intro i. etrans. { apply pathsinv0, bifunctor_equalwhiskers. } apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import  UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.ProductCategory. Import BifunctorNotations. Import BifunctorNotations. Section FunctorsFromProductCategory. Import BifunctorNotations. End FunctorsFromProductCategory. Import BifunctorNotations. Import BifunctorNotations. Import BifunctorNotations.