Lemma coprod_antidistributor_nat (cs1 cs2 : power_category I C) (g : power_category I C ⟦ cs1, cs2 ⟧) : coprod_antidistributor cs1 · #F (#(coproduct_functor I CPC) g) = #(coproduct_functor I CPD) (#(family_functor I (fun _ => F)) g) · coprod_antidistributor cs2. Proof. etrans. { apply postcompWithCoproductArrow. } etrans. 2: { apply pathsinv0, precompWithCoproductArrow. } apply maponpaths. apply funextsec; intro i. etrans. { apply pathsinv0, functor_comp. } etrans. 2: { cbn. apply functor_comp. } apply maponpaths. apply CoproductInCommutes. 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.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.ProductCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.whiskering.