  We show that any completely positive multiplier of the convolution algebra of
the dual of an operator algebraic quantum group $\G$ (either a locally compact
quantum group, or a quantum group coming from a modular or manageable
multiplicative unitary) is induced in a canonical fashion by a unitary
corepresentation of $\G$. It follows that there is an order bijection between
the completely positive multipliers of $L^1(\G)$ and the positive functionals
on the universal quantum group $C_0^u(\G)$. We provide a direct link between
the Junge, Neufang, Ruan representation result and the representing element of
a multiplier, and use this to show that their representation map is always
weak$^*$-weak$^*$-continuous.
