Proposition 58.8.4. Let $f : X \to Y$ be a universal homeomorphism of schemes. Then

$\textit{FÉt}_ Y \longrightarrow \textit{FÉt}_ X,\quad V \longmapsto V \times _ Y X$

is an equivalence. Thus if $X$ and $Y$ are connected, then $f$ induces an isomorphism $\pi _1(X, \overline{x}) \to \pi _1(Y, \overline{y})$ of fundamental groups.

Proof. Recall that a universal homeomorphism is the same thing as an integral, universally injective, surjective morphism, see Morphisms, Lemma 29.45.5. In particular, the diagonal $\Delta : X \to X \times _ Y X$ is a thickening by Morphisms, Lemma 29.10.2. Thus by Lemma 58.8.3 we see that given a finite étale morphism $U \to X$ there is a unique isomorphism

$\varphi : U \times _ Y X \to X \times _ Y U$

of schemes finite étale over $X \times _ Y X$ which pulls back under $\Delta$ to $\text{id} : U \to U$ over $X$. Since $X \to X \times _ Y X \times _ Y X$ is a thickening as well (it is bijective and a closed immersion) we conclude that $(U, \varphi )$ is a descent datum relative to $X/Y$. By Étale Morphisms, Proposition 41.20.6 we conclude that $U = X \times _ Y V$ for some $V \to Y$ quasi-compact, separated, and étale. We omit the proof that $V \to Y$ is finite (hints: the morphism $U \to V$ is surjective and $U \to Y$ is integral). We conclude that $\textit{FÉt}_ Y \to \textit{FÉt}_ X$ is essentially surjective.

Arguing in the same manner as above we see that given $V_1 \to Y$ and $V_2 \to Y$ in $\textit{FÉt}_ Y$ any morphism $a : X \times _ Y V_1 \to X \times _ Y V_2$ over $X$ is compatible with the canonical descent data. Thus $a$ descends to a morphism $V_1 \to V_2$ over $Y$ by Étale Morphisms, Lemma 41.20.3. $\square$

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