  We prove that for every digraph $D$ and every choice of positive integers
$k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together
with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for
every digraph $C$ of order at most $k$, there exists an acyclic homomorphism
$D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and
(ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic
homomorphism $\varphi\colon D^*\to C$ there exists a unique acyclic
homomorphism $f\colon D\to C$ such that $\varphi=f\circ\psi$. This implies the
main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with
large girth, Canad. J. Math., 64(6) (2012), 1310-1328; MR2994666] analogously
with how the work [J. Ne\v{s}et\v{r}il and X. Zhu, On sparse graphs with given
colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161-172;
MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with
large girth, J. Graph Theory, 23(1) (1996), 33-41; MR1402136].
