  On a Fock space constructed from $mn$ free bosons and lattice ${\Bbb
{Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra
$\mathscr {E}_m$ associated to $\mathfrak{gl}_m$, together with a level $m$
action of the quantum toroidal algebra ${\mathscr E}_n$ associated to
${\mathfrak {gl}}_n$. We prove that the $\mathscr {E}_m$ transfer matrices
commute with the $\mathscr {E}_n$ transfer matrices after an appropriate
identification of parameters.
