Lift functor from Set to Stone

Question

Setting: Let $\mathbb{T}$ be an endofunctor on $\mathsf{Set}$, the category of sets and functions. Let $\mathsf{Stone}$ be the category of Stone spaces and continuous functions. In [1] (in the conclusion) the authors write: "If $\mathbb{T}$ preserves finite sets, then there is a canonical lifting from $\mathsf{Set}$ to $\mathsf{Stone}$."

Question: I am looking for a reference for (or explanation of) this canonical way to lift a functor.

Here [1] is the article Ultrafilter Extensions for Coalgebras by Kupke, Kurz and Pattinson (2005).

Answer 1

I'm not totally sure what Kupke, Kurz, and Pattinson are referring to here, because I don't know what exactly is required to call a functor $\mathsf{Stone}\to\mathsf{Stone}$ a "lift" of a functor $\mathsf{Set}\to\mathsf{Set}$. But here's a guess:

$\mathsf{Set}$ and $\mathsf{Stone}$ have a common full subcategory, $\mathsf{FinSet}$. Moreover, $\mathsf{Set}$ is the ind-completion of $\mathsf{FinSet}$, and $\mathsf{Stone}$ is its pro-completion. Essentially, this means that every set is canonically a directed colimit of finite sets, and every Stone space is canonically a codirected limit of finite sets.

Then a functor $T\colon \mathsf{FinSet}\to \mathsf{FinSet}$ induces a functor $T_\mathrm{ind}\colon \mathsf{Set}\to \mathsf{Set}$, by $T_\mathrm{ind}(X) = \varinjlim T(Y)$ where the colimit is over the image under $T$ of the full diagram of all finite subsets $Y\hookrightarrow X$, as well as a functor $T_\mathrm{pro}\colon \mathsf{Stone}\to\mathsf{Stone}$, by $T_\mathrm{pro}(X) = \varprojlim T(Y)$, where the limit is over the image under $T$ of the full diagram of all finite quotients $X\twoheadrightarrow Y$.

So if you have a functor $T\colon \mathsf{Set}\to \mathsf{Set}$ which preserves finite sets, a natural way to get a functor $\widehat{T}\colon \mathsf{Stone}\to \mathsf{Stone}$ is by restricting and then extending: $\widehat{T} = (T|_{\mathsf{FinSet}})_\mathrm{pro}$.

If we define $\widehat{T}$ in this way, there is a natural transformation $\beta \circ T\to \widehat{T}\circ \beta$, and by adjointness also $T\to U\circ \widehat{T}\circ \beta$. Maybe this is the sense in which it's a lift?