How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable? Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module spectrum in the infinity category of cyclotomic spectra. Does this follow from the fact that a smooth proper stable infinity category is dualizable in the category $\text{Cat}^{\text{perf}}_{\infty,k}$ ?
 A: $\newcommand{\THH}{\mathrm{THH}}
\newcommand{\Cat}{\mathrm{Cat}}
\newcommand{\perf}{\mathrm{perf}}
\newcommand{\Sp}{\mathrm{Sp}}
\newcommand{\Mod}{\mathrm{Mod}}$
If you ask about dualizability in $\THH(k)$-modules in $\Sp$, it indeed follows from this together with the fact that $\THH: \Cat^{\perf}_{\infty,k}\to \Mod_{\THH(k)}(\Sp)$ is symmetric monoidal, hence preserves dualizability.
This symmetric monoidality follows from the following two properties of $\THH: \Cat^{\perf}_\infty\to \Sp$:
1- It is symmetric monoidal
2- It commutes with sifted colimits.
The latter ensures that if you do a relative tensor product (which is defined via a bar construction, which is a colimit of a simplicial object), you can do it before or after applying $\THH$.
The exact same argument works for modules over $\THH(k)$ in $\Sp^{BS^1}$.
If you ask, as you did, about modules in cyclotomic spectra, then this is still so, and this is still because $\THH$ has these two properties with values in cyclotomic spectra, but it is somehow not as easy to prove : for starters, I don't know of a convenient reference for $\Cat^\perf_\infty\to \mathrm{CycSp}$, although it is folklore that such a thing exists and is symmetric monoidal.
Once you have this, the rest follows in exactly the same way because colimits in $\mathrm{CycSp}$ are computed underlying, say, by corollary II.1.7. in Nikolaus-Scholze.
