  The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by
Chartrand et al., is a natural and nice generalization of the concept of
(vertex-)connectivity. In this paper, we prove that for any two connected
graphs $G$ and $H$, $\kappa_3(G\circ H)\geq \kappa_3(G)|V(H)|$. We also give
upper bounds for $\kappa_3(G\Box H)$ and $\kappa_3(G\circ H)$. Moreover, all
the bounds are sharp.
