Recently I became through [this mathoverflow question][1] aware of the article [Codensity and the ultrafilter monad][2] by Tom Leinster. There he shows that the ultrafiler monad on the category $\mathrm{Set}$ arises from the adjunction
$$ \mathrm{Set} \rightleftarrows \mathrm{Fun}(\mathrm{FinSet}, \mathrm{Set})^{\mathrm{op}},$$
where the left adjoint is given by the coYoneda-embedding (that it has a right adjoint follows either by a construction or the adjoint functor theorem). Moreover it is known that the category of compact Hausdorff spaces is monadic over $\mathrm{Set}$ and that the corresponding monad is the ultrafilter monad as well, exhibiting the category of compact Hausdorff spaces as algebras over this monad. 

Moving to $\infty$-categories, it is natural to replace $\mathrm{Set}$ by the $\infty$-category $\mathcal{S}$ of spaces (or animae, as some call it). This has the sub-$\infty$-category $\mathcal{S}^{\mathrm{fin}}$ of finite spaces (i.e. the smallest finitely cocomplete subcategory containing the point). Using the coYoneda embedding and the adjoint functor theorem, we obtain again an adjunction
$$\mathcal{S}\rightleftarrows \mathrm{Fun}(\mathcal{S}^{\mathrm{fin}}, \mathcal{S})^{\mathrm{op}}.$$
Can one describe the resulting monad and algebras over it? Is it a known $\infty$-category? Moreover, one might ask about its relation to to other $\infty$-categories, like profinite spaces or condensed spaces. 



  [1]: https://mathoverflow.net/questions/104777/what-are-the-algebras-for-the-double-dualization-monad
  [2]: https://arxiv.org/abs/1209.3606