Dualizable presentable $\mathrm{\infty }$-categories

We prove that for any presentably symmetric monoidal $\mathrm{\infty }$-category $\mathcal{V}$, the $\mathrm{\infty }$-category ${\mathbf{M}\mathbf{o}\mathbf{d}}_{\mathcal{V}}({\mathbf{P}\mathbf{r}}^{\mathrm{L}}{)}^{\mathrm{d}\mathrm{b}\mathrm{l}}$ of dualizable presentable $\mathcal{V}$-modules and internal left adjoints between them is itself presentable. Along the way, we survey formal properties of these dualizable $\mathcal{V}$-modules. We pay close attention to the case of the $\mathrm{\infty }$-category of spectra, where we survey the foundational properties of ``compact morphisms’'.