  Cartesian product networks are always regarded as a tool for ``combining''
two given networks with established properties to obtain a new one that
inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq
V(F)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner
tree connecting $S$} (or simply, \emph{an $S$-tree}) is a subgraph $T=(V',E')$
of $F$ that is a tree with $S\subseteq V'$. For $S\subseteq V(F)$ and $|S|\geq
2$, the {\it generalized local edge-connectivity} $\lambda(S)$ is the maximum
number of edge-disjoint Steiner trees connecting $S$ in $F$. For an integer $k$
with $2\leq k\leq n$, the {\it generalized $k$-edge-connectivity}
$\lambda_k(F)$ of a graph $F$ is defined as
$\lambda_k(F)=\min\{\lambda(S)\,|\,S\subseteq V(F) \ and \ |S|=k\}$.In this
paper, we give sharp upper and lower bounds for $\lambda_k(G\Box H)$, where
$\Box$ is the Cartesian product operation, and $G,H$ are two graphs.
