The Stacks project

Proof. Let $U \to X$ be an étale morphism such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale and $U \to V$ quasi-compact with geometrically connected fibres. Observe that $U \to V$ is flat (More on Flatness, Lemma 38.2.3). We claim that

Namely, thinking of $U$ as an object of $X_{\acute{e}tale}$ and $V$ as an object of $S_{\acute{e}tale}$ we see that the first equality follows from Lemma 59.39.3¹. Thinking of $V \times _ S T$ as an object of $T_{\acute{e}tale}$ the second equality follows from the definition of $g_*$. Observe that $U \times _ X Y = U \times _ S T$ (because $Y = X \times _ S T$) and hence $U \times _ X Y \to V \times _ S T$ has geometrically connected fibres as a base change of $U \to V$. Thinking of $U \times _ X Y$ as an object of $Y_{\acute{e}tale}$, we see that the third equality follows from Lemma 59.39.3 as before. Finally, the fourth equality follows from the definition of $h_*$.

Since by assumption every object of $X_{\acute{e}tale}$ has an étale covering to which the argument of the previous paragraph applies we see that the lemma is true. $\square$