The Stacks project

Proof. We may think of $y$ as a morphism $U \to \mathcal{Y}$. By Properties of Stacks, Lemma 100.5.3 and Lemmas 101.27.3 and 101.25.3 we see that $\mathcal{X} \times _\mathcal {Y} U \to U$ is surjective, flat, and locally of finite presentation. Let $V$ be a scheme and let $V \to \mathcal{X} \times _\mathcal {Y} U$ smooth and surjective. Then $V \to \mathcal{X} \times _\mathcal {Y} U$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas 67.37.7 and 67.37.5). Hence also $V \to U$ is surjective, flat, and locally of finite presentation, see Properties of Stacks, Lemma 100.5.2 and Lemmas 101.27.2, and 101.25.2. Hence $\{ V \to U\}$ is the desired fppf covering and $x : V \to \mathcal{X}$ is the desired object. $\square$