The "ultrafilter monad" $X\mapsto \mathrm{U}(X)$ maps a set $X$ to the set of ultrafilters on it. The abstractness of the characterization of compact Hausdorff spaces lies in the fact that $\mathrm{U}$ is defined in purely set-theoretical (or, rather, category-theoretical) terms: it appears to be the "codensity monad" (don't ask me the meaning of this because I don't know!) of the inclusion $\mathrm{FinSet}\to\mathrm{Set}$.