$\xymatrix{ X \ar[d] & X' \ar[l] \ar[d] \\ S & S' \ar[l] }$

be a commutative diagram of schemes with $X' \to X$ and $S' \to S$ étale. Let $s' \in S'$ be a point. Then

$X' \times _{S'} \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \longrightarrow X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$

is étale.

Proof. This is true because $X' \to X_{S'}$ is étale as a morphism of schemes étale over $X$, see Morphisms, Lemma 29.36.18 and the base change of an étale morphism is étale, see Morphisms, Lemma 29.36.4. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).