Duals and adjoints in higher Morita categories

We study duals for objects and adjoints for $k$-morphisms in
$\operatorname{Alg}_n(\mathcal{S})$, an $(\infty,n+N)$-category that models a
higher Morita category for $E_n$ algebra objects in a symmetric monoidal
$(\infty,N)$-category $\mathcal{S}$. Our model of
$\operatorname{Alg}(\mathcal{S})$ uses the geometrically convenient framework
of factorization algebras. The main result is that
$\operatorname{Alg}_n(\mathcal{S})$ is fully $n$-dualizable, verifying a
conjecture of Lurie. Moreover, we unpack the consequences for a natural class
of fully extended topological field theories and explore $(n+1)$-dualizability.