The Stacks project

Proof. Recall that $U$ gives rise to a representable sheaf $\mathcal{F} = h_ U$ in $\mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$, see Properties of Spaces, Section 66.18. The descent datum on $U$ relative to $f$ exactly gives a descent datum $(\mathcal{F}, \varphi )$ for étale sheaves with respect to $\{ X \to Y\} $. By Lemma 81.4.5 this descent datum is effective. Let $\mathcal{G}$ be the corresponding sheaf on $Y_{\acute{e}tale}$. By Properties of Spaces, Lemma 66.27.3 we obtain an étale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{G}$; we omit the verification of the set theoretic condition¹. The given isomorphism $\mathcal{F} \to f_{small}^{-1}\mathcal{G}$ corresponds to an isomorphism $U \to V \times _ Y X$ compatible with the descent datum. $\square$