In Milne's notes on 'Etale Cohomology, Proposition 6.16 on page 49 is the following:
Proposition 6.16 Assume $X$ is connected, and let $\overline{x}$ be a geometric point of $X$. The map $\mathscr{F}\mapsto \mathscr{F}_{\overline{x}}$ defines an equivalence between the category of locally constant sheaves of sets (resp. abelian groups) with finite stalks on $X$ and the category of finite $\pi_1(X,\overline{x})$-sets (resp. modules).
In the proof, one has a surjective 'etale morphism $Z\to X$. Then, one considers a surjective finite etale map $Z'\to Z$ such that $Z'\to X$ is a Galois covering. It is then claimed that $Z\times_X Z'$ is a disjoint union of copies of $Z'$. I don't quite see how this follows, and I'd be grateful if someone could elaborate what exactly happened.
Thanks.
Milne's definition of Galois covering applied to the map $Z' \to X$ says that there is a group $G$ acting on $Z'$ on the right such that the map $Z' \times G \to Z' \times_X Z'$ given by $(z, g) \mapsto (z, zg)$ is an isomorphism. Composing with the surjection $Z' \times_X Z' \to Z' \times_X Z$ induced by the surjection $Z' \to Z$, we get a surjection $$Z' \times G \to Z' \times_X Z$$ which is the identity on the left factor. This exactly realizes $Z' \times_X Z$ as a disjoint union of copies of $Z'$ (with the copies indexed by the coset space $G/H$, where $H$ is the subgroup that acts as the identity on $Z$).