Let C be a finite category. For an object X of C one has the hom-functor Hom(-,X) of C to Set. If G is a subgroup of Aut(X), one has the quotient functor Hom(-,X)/G. We show that any finite product of hom-functors of C is a sum of hom-functors if and only if C has pushouts and coequalizers and that any finite product of hom-functors of C is a sum of functors of the form \Hom(-,X)/G if and only if C has pushouts. These are variations of the fact that a finite category has products if and only if it has coproducts.
Keywords: pushout, coequalizer, hom-functor, familially representable functor, nearly representable functor
2010 MSC: 18A30, 18A35
Theory and Applications of Categories, Vol. 30, 2015, No. 30, pp 1017-1031.