  Koh and Tay proved a fundamental classification of $G$ vertex-multiplications
into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. In this
paper, we prove that vertex-multiplications of cartesian products of graphs
$G\times H$ lie in $\mathscr{C}_0$ ($\mathscr{C}_0\cup \mathscr{C}_1$ resp.) if
$G^{(2)}\in \mathscr{C}_0$ ($\mathscr{C}_1$ resp.), $d(G)\ge 2$ and $d(G\times
H)\ge 4$. We also focus on cartesian products involving trees, paths and cycles
and show that most of them lie in $\mathscr{C}_0$.
