This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces

To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\text{CHaus}$ of compact Hausdorff spaces in a nice way. Every compact Hausdorff space $X$ can be covered by a profinite set (specifically the stone Czech compactification of the underlying set of $X$). $\text{Prof}$ can be characterized by a universal property without reference to the category of topological spaces: It is the free completion of the category $\text{Fin}$ of finite sets under cofiltered limits.

Since $\text{CHaus}$ is, unlike $\text{Top}$, a category that is quite nicely behaved (it is for example the category of algebras for the ultrafilter monad on sets), it seems natural to ask: Can it also be characterized as a category by a universal property similar to $\text{Prof}$?

A Characterisation of the Category of Compact Hausdorff Spaces, Theory App. Cat.35, (2020), pp.1871–1906 (arxiv version). The authors show that $CHaus$ is, up to equivalence, "the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object". Refer to $\S4$ of the paper for their notion of filtrality. $\endgroup$1more comment