  As a quantum affinization, the quantum toroidal algebra is defined in terms
of its "left" and "right" halves, which both admit shuffle algebra
presentations. In the present paper, we take an orthogonal viewpoint, and give
shuffle algebra presentations for the "top" and "bottom" halves instead,
starting from the evaluation representation of the quantum affine group and its
usual R-matrix. An upshot of this construction is a new topological coproduct
on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on
the horizontal quantum affine subalgebra.
