We fully characterise the situations where the existence of a homomorphism from a digraph
to at least one of a finite set
of directed graphs is determined by a finite number of forbidden subgraphs. We prove that these situations, called
, are characterised by the non-existence of a homomorphism to
from a finite set of forests.
Furthermore, we characterise all finite maximal antichains in the partial order of directed graphs ordered by the existence of homomorphism. We show that these antichains correspond exactly to the generalised dualities. This solves a problem posed in . Finally, we show that it is NP-hard to decide whether a finite set of digraphs forms a maximal antichain.