The Stacks project

Remark 59.45.3. In the situation of Theorem 59.45.2 it is also true that $V \mapsto V_ X$ induces an equivalence between those étale morphisms $V \to Y$ with $V$ affine and those étale morphisms $U \to X$ with $U$ affine. This follows for example from Limits, Proposition 32.11.2.