  We prove that two dual operator algebras are weak$^*$ Morita equivalent if
and only if they have equivalent categories of dual operator modules via
completely contractive functors which are also weak$^*$-continuous on
appropriate morphism spaces. Moreover, in a fashion similar to the operator
algebra case, we characterize such functors as the module normal Haagerup
tensor product with an appropriate weak$^*$ Morita equivalence bimodule. We
also develop the theory of the $W^*$-dilation, which connects the
non-selfadjoint dual operator algebra with the $W^*$-algebraic framework. In
the case of weak$^*$ Morita equivalence, this $W^*$-dilation is a $W^*$-module
over a von Neumann algebra generated by the non-selfadjoint dual operator
algebra. The theory of the $W^*$-dilation is a key part of the proof of our
main theorem.
