  For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping
$f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each
edge $uv$ of $G$. The smallest integer $k$ for which there is a proper
$k$-coloring of $G$ is called the chromatic number of $G$, denoted by
$\chi(G)$. A locally identifying coloring (for short, lid-coloring) of a graph
$G$ is a proper $k$-coloring of $G$ such that every pair of adjacent vertices
with distinct closed neighborhoods has distinct set of colors in their closed
neighborhoods. The smallest integer $k$ such that $G$ has a lid-coloring with
$k$ colors is called locally identifying chromatic number (for short,
lid-chromatic number) of $G$, denoted by $\chi_{lid}(G)$. This paper studies
the lid-coloring of the Cartesian product and tensor product of two graphs. We
prove that if $G$ and $H$ are two connected graphs having at least two vertices
then (a) $\chi_{lid}(G \square H) \leq \chi(G) \chi(H)-1$ and (b) $\chi_{lid}(G
\times H) \leq \chi(G) \chi(H)$. Here $G \square H$ and $G \times H$ denote the
Cartesian and tensor products of $G$ and $H$ respectively. We determine the
lid-chromatic number of $C_m \square P_n$, $C_m \square C_n$, $P_m \times P_n$,
$C_m \times P_n$ and $C_m \times C_n$, where $C_m$ and $P_n$ denote a cycle and
a path on $m$ and $n$ vertices respectively.
