  An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of $l$ colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such
that $V(F)=V$ for every $F\in \Gamma$. Consider any function
$h:E(D)\longrightarrow\Gamma $. Then the product $D\otimes_{h} \Gamma$ is the
digraph with vertex set $V(D)\times V$ and $((a,x),(b,y))\in
E(D\otimes_{h}\Gamma)$ if and only if $ (a,b)\in E(D)$ and $ (x,y)\in E(h
(a,b))$.
  In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the $\otimes_h$-product of oriented cycles. We study the
behavior of the $\otimes_h$-product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.
