The ultrafilter-limit map of a compact Hausdorff space is an algebra for the ultrafilter monad

By [L1], ${\xi }_X$ is defined on every ultrafilter. When $X=\mathrm{\varnothing }$, both $\beta X$ and $X$ are empty and the unique empty map satisfies the equations below.

The principal ultrafilter ${\eta }_X(x)$ contains every neighbourhood of $x$, so it converges to $x$. Uniqueness in [L1] gives ${\xi }_X{\eta }_X(x)=x$ for every $x$.

Let $\mathbb{W}$ be an ultrafilter on $\beta X$. If an open neighbourhood $O$ of a point belongs to the pushforward $\beta ({\xi }_X)(\mathbb{W})$, then ${\xi }_X^{-1}[O]\in \mathbb{W}$. Every ultrafilter whose limit lies in $O$ contains $O$, so ${\xi }_X^{-1}[O]\subseteq \widehat{O}$; upward closure gives $\widehat{O}\in \mathbb{W}$, hence $O\in {\mu }_X(\mathbb{W})$ by [L2]. Thus every limit of the pushforward is a limit of the flattening.

Both ultrafilters in step 2.2 have unique limits by [L1], so their limits coincide: ${\xi }_X\beta ({\xi }_X)(\mathbb{W})={\xi }_X{\mu }_X(\mathbb{W})$.

Steps 2.1 and 3.1 are exactly the unit and multiplication equations in [L3], so ${\xi }_X$ is an ultrafilter algebra.