  We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is
exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous
$\infty$-category. The heart of the proof is the description of the
$\infty$-category of $\mathcal C$-valued sheaves on $\mathcal X$ as an
$\infty$-category of functors that satisfy finite limits conditions as well as
filtered colimits conditions (instead of limits conditions purely); we call
such functors $\omega$-continuous sheaves.
  As an application, we show that when $\mathcal X$ is exponentiable, its
$\infty$-category of stable sheaves $\mathcal S\mathrm{h}(\mathcal X,
\mathrm{Sp})$ is a dualisable object in the $\infty$-category of presentable
stable $\infty$-categories.
