proposition: Lemma coprod_antidistributor_nat_left (a : A) (cs1 cs2 : power_category I C) (g : power_category I C ⟦ cs1, cs2 ⟧) : bifunctor_coprod_antidistributor a cs1 · a ⊗^{F}_{l} #(coproduct_functor I CPC) g = #(coproduct_functor I CPD) (#(family_functor I (fun _ => leftwhiskering_functor F a)) g) · bifunctor_coprod_antidistributor a cs2.
proof: Proof. etrans. { apply postcompWithCoproductArrow. } etrans. 2: { apply pathsinv0, precompWithCoproductArrow. } apply maponpaths. apply funextsec; intro i. etrans. { apply pathsinv0, (functor_comp (leftwhiskering_functor F a)). } etrans. 2: { cbn. apply (functor_comp (leftwhiskering_functor F a)). } apply maponpaths. apply CoproductInCommutes. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.ProductCategory