A finite braided tensor category is fully dualizable in the Morita 4-category of braided pre-tensor categories whenever its symmetric center is separable.
Some remarks on higher Morita categories
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abstract
We take another look at the construction of double $\infty$-categories of algebras and bimodules and prove a few supplemental results about these, including a simpler proof of the Segal condition and a comparison between our construction and that of Lurie. We then take a more streamlined, inductive approach to the higher Morita categories of $E_{n}$-algebras and show that these deloop correctly.
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Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories
A finite braided tensor category is fully dualizable in the Morita 4-category of braided pre-tensor categories whenever its symmetric center is separable.