  Consider a finite simple digraph $D$ with vertex set $V(D)$. An Italian
dominating function (IDF) on $D$ is a function $f:V(D)\rightarrow\{0,1,2\}$
satisfying every vertex $u$ with $f(u)=0$ has an in-neighbor $v$ with $f(v)=2$
or two in-neighbors $w$ and $z$ with $f(w)=f(z)=1$. A total Italian dominating
function (TIDF) on $D$ is an IDF $f$ such that the subdigraph $D[\{ u\, |\,
f(u)\ge 1\}]$ contains no isolated vertices. The weight $\omega(f)$ of a TIDF
$f$ on $D$ is $\sum_{u\in V(D)}f(u)$. The total Italian domination number of
$D$ is $\gamma_{tI}(D)=\min\{ \omega(f)\, |\, \mbox{$f$ is a TIDF on $D$}\}$.
  In this paper, we present bounds on $\gamma_{tI}(D)$, and investigate the
relationship between several different domination parameters. In particular, we
give the total Italian domination number of the Cartesian products $P_2\Box
P_n$ and $P_3\Box P_n$, where $P_n$ represents a dipath with $n$ vertices.
