Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter 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.

Edit: As Denis and Dustin pointed out, it is much more natural to replace $\mathrm{FinSet}$ by the $\infty$-category of $\pi$-finite spaces (instead of $\mathcal{S}^{\mathrm{fin}}$), i.e. spaces whose homotopy groups are concentrated in finitely many degrees and are finite there.