proposition: Lemma functor_to_bifunctor_to_functor {C D E : category} (F : functor (C × D) E) : bifunctor_to_functorfromproductcat (bifunctor_from_functorfromproductcat F) = F.
proof: Proof. apply (functor_eq _ _ (homset_property _)). use total2_paths_f. - apply idpath. - etrans. { rewrite idpath_transportf. apply idpath. } repeat (apply funextsec ; intro). etrans. { apply (! functor_comp _ _ _). } apply maponpaths. etrans. { apply binprod_comp. } cbn. etrans. { apply maponpaths ; apply id_left. } etrans. { apply maponpaths_2 ; apply id_right. } apply idpath. 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