gn.general topology - profinite spaces are the pro-completion of finite sets

Let $f:X to Z$ be a map to a finite discrete space. Note that each fiber, $f^{-1}(z)$, is both open and closed in $X$. Let $p_i: X to X_i$ be the projection maps.

Fix some $z in Z$. Since $f^{-1}(z)$ is open, and the fibers of the maps are a basis, there is an open cover of $f^{-1}(z)$ by sets of the form $p_i^{-1}(x)$, for $x$ in various $X_i$.
Since $f^{-1}(z)$ is closed in a compact space, it is compact. So we can take a finite subcover of this cover. Thus, there is some single index $i$ for which $f^{-1}(z)$ is covered by sets of the form $p_i^{-1}(x)$, $x in X_i$.

Since $Z$ is finite, there is a single $i$ such that, for every $z in Z$, the fiber $f^{-1}(z)$ is covered by sets of the form $p_i^{-1}(x)$, $x in X_i$. The map $f$ factors through $X_i$.