  The dichromatic number of a digraph $G$ is the smallest integer $\chi_a(G)$
such that the vertex set of $G$ can be partitioned into $\chi_a(G)$ sets, each
of which induces an acyclic subdigraph. This is a generalization of the classic
chromatic number of graphs. Here, we investigate the dichromatic number of the
cartesian, direct, strong and lexicographic products, giving generalizations of
some classic results on the chromatic number of products. More specifically, we
prove that the following inequalities, known to hold for the chromatic number
of graphs, still hold for the dichromatic number of digraphs: $\chi_a(G\square
H)=\max\{\chi_a(G),\chi_a(H)\}$; $\chi_a(G\times H)\le
\min\{\chi_a(G),\chi_a(H)\}$; and $\chi_a(G[H]) =
\chi_a(G[\overset{\leftrightarrow}{K}_k])$, where $k =\chi_a(H)$ and
$\overset{\leftrightarrow}{K}_k$ denotes the complete digraph on $k$ vertices.
In addition, we investigate the products of directed cycles, giving exact
values for $\chi_a(\overset{\rightarrow}{C}_n\times
\overset{\rightarrow}{C}_m)$ and $\chi_a(\overset{\rightarrow}{C}_n\boxtimes
\overset{\rightarrow}{C}_m)$ for every $n,m$, and for
$\chi_a(\overset{\rightarrow}{C}_n[H])$ for every positive integer $n$. This
latter result generalizes a result given in \cite{PP.16}, where they give exact
values when $n>\chi_a(H)$. We also provide a upper-bound to the dichromatic
number of a digraph $G$ as a function of the treewidth of its underlying graph
and we present an {\FPT}-time algorithm that computes the dichromatic number of
$G$, when parameterized by treewidth of the underlying graph of $G$.
