  In contrast to the abundance of "direct" Ramsey results for classes of finite
structures (such as finite ordered graphs, finite ordered metric spaces and
finite posets with a linear extension), in only a handful of cases we have a
meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for
finite ordered oriented graphs. Instead of embeddings, which are crucial for
"direct" Ramsey results, we consider a special class of surjective
homomorphisms between finite ordered oriented graphs. Since the setting we are
interested in involves both structures and morphisms, all our results are
spelled out using the reinterpretation of the (dual) Ramsey property in the
language of category theory.
