Brown functors of directed graphs

We prove that any digraph Brown functor – i.e. a contravariant functor from the homotopy category of finite directed graphs to the category of abelian groups, satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris axiom – is representable. Furthermore, we show that the first path cohomology functor is a digraph Brown functor.