The Stacks project

Proof. After base changing via $Y \to S$, we may assume that $Y = S$. Thus we may and do assume both $X$ and $Y$ are étale over $S$. In other words, the theorem states that the base change functor is a fully faithful functor from the category of schemes étale over $S$ to the category of schemes étale over $S'$.

Consider the forgetful functor

We claim this functor is an equivalence. On the other hand, the functor

is fully faithful by Étale Morphisms, Lemma 41.20.3. Thus the claim implies the theorem.

Proof of the claim. 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 $\Delta : S' \to S' \times _ S S'$ is a thickening by Morphisms, Lemma 29.10.2. Thus by Étale Morphisms, Theorem 41.15.1 we see that given $X' \to S'$ étale there is a unique isomorphism

of schemes étale over $S' \times _ S S'$ which pulls back under $\Delta $ to $\text{id} : X' \to X'$ over $S'$. Since $S' \to S' \times _ S S' \times _ S S'$ is a thickening as well (it is bijective and a closed immersion) we conclude that $(X', \varphi ')$ is a descent datum relative to $S'/S$. The canonical nature of the construction of $\varphi '$ shows that it is compatible with morphisms between schemes étale over $S'$. In other words, we obtain a quasi-inverse $X' \mapsto (X', \varphi ')$ of the functor (59.45.1.1). This proves the claim and finishes the proof of the theorem. $\square$