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Abstract. We prove that two dual operator algebras are weak ∗ Morita equivalent in the sense of [4] 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

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is to study its category of representations

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