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Let 
$C$ and 
$D$ be digraphs. A mapping 
$f:V\left( D \right)\to V\left( C \right)$ is a $C$-colouring if for every arc 
$uv$ of $D$, either 
$f\left( u \right)f\left( v \right)$ is an arc of $C$ or 
$f\left( u \right)=f\left( v \right)$, and the preimage of every vertex of $C$ induces an acyclic subdigraph in $D$. We say that $D$ is $C$-colourable if it admits a $C$-colouring and that $D$ is uniquely $C$-colourable if it is surjectively $C$-colourable and any two $C$-colourings of $D$ differ by an automorphism of $C$. We prove that if a digraph $D$ is not $C$-colourable, then there exist digraphs of arbitrarily large girth that are $D$-colourable but not $C$-colourable. Moreover, for every digraph $D$ that is uniquely $D$-colourable, there exists a uniquely $D$-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number 
$r\ge 1$, there are uniquely circularly 
$r$-colourable digraphs with arbitrarily large girth.
