Let $(H,r)$ be a fixed rooted digraph. The $(H,r)$-coloring problem is the problem of deciding for which rooted digraphs $(G,s)$ there is a homomorphism $f:G\to H$ which maps the vertex $s$ to the vertex $r$. Let $(H,r)$ be a rooted oriented path. In this case we characterize the nonexistence of such a homomorphism by the existence of a rooted oriented cycle $(C,q)$, which is homomorphic to $(G,s)$ but not homomorphic to $(H,r)$. Such a property of the digraph $(H,r)$ is called rooted cycle duality or $*$-cycle duality. This extends the analogical result for...