An isomorphism theorem for digraphs

A seminal result by Lovász states that two digraphs A and B (possibly with loops) are isomorphic if and only if for every digraph X the number of homomorphisms X → A equals the number of homomorphisms X → B. Lovász used this result to deduce certain cancellation properties for the direct product of digraphs. We develop an analogous result for the class of digraphs without loops, and with weak homomorphisms replacing homomorphisms. We show that two digraphs A and B (without loops) are isomorphic if and only if the number of weak homomorphisms X → A equals the number of weak homomorphisms X → B. This result is then applied to deduce a general cancellation property for the strong product of digraphs as well as graphs.