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This is a follow up to this MO question: Free symmetric monoidal $(\infty,n)$-categories with duals

Freed-Hopkins-Lurie-Teleman define a classical field theory as a symmetric monoidal functor $I$ from $n$-cobordism to the symmetric monoidal $n$-category $Fam_n(\mathcal{C})$ of $n$-families over a fixed symmetric monoidal $n$-category. This is required to lift a certain "geometric background", i.e., a fixed functor from $n$-cobordism to the $n$-category $Fam_n$.

If $\mathcal{C}$ in an $(\infty,n)$ symmetric monoidal $n$-category with duals, then it is likely that $Fam_n(\mathcal{C})$ has duals, too (I have so far checked this only for $n=1$; in this particular case, the dual of a functor $X\to \mathcal{C}$ from a finite groupoid $X$ to $\mathcal{C}$ is nothing but the dual representation of $X$; and I expect this will still be true for higher $n$'s). Then, by the cobordism hypothesis, the datum of $I$ would be reduced to the choice of a fully dualizable object $I(*)$ in $Fam_n(\mathcal{C})$, i.e., of a functor $X\to \mathcal{C}$ "with a few good properties". The object $I(*)$ is explicited a couple of times in the paper (namely in the cases $n=1$ and $n=2$), but its fully dualizability is never discussed, nor seems to be claimed that $I(*)$ already contains all the information of $I$. For $n=3$, instead, the natural 3-representation of the groupoid $*//G$ which would starightforwardly generalize the $n=1$ and $2$ cases is not considered explicitely, and only its integrated (or quantized) version apperas.

Therefore, here is the question:

Is it correct that a classical field theory $I$ in the sense of Freed-Hopkins-Lurie-Teleman are equivalent to the datum of a fully dualizable object $I(*):X\to \mathcal{C}$ in $Fam_n(\mathcal{C})$ as claimed above? (with the "background geometry" given by the groupoid $X$)

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