  We prove that the quantum toroidal algebras $\mathcal{E}_\mathbf{s}$
associated with different root systems $\mathbf{s}$ of $\mathfrak{gl}_{m|n}$
type are isomorphic. We also show the existence of Miki automorphism of
$\mathcal{E}_\mathbf{s}$, which exchanges the vertical and horizontal
subalgebras.
  To obtain these results, we establish an action of the toroidal braid group
on the direct sum $\oplus_\mathbf{s} \mathcal{E}_\mathbf{s}$ of all such
algebras.
