The Stacks project

Proof. We may apply the result of Lemma 106.7.4. Thus it suffices to construct a smooth surjective morphism $W' \to T'$ with $W'$ affine such that $x|_{T \times _{W'} T'}$ lifts to $W'$. (We urge the reader to find their own proof of this fact using the analogous result for algebraic spaces already established.) We choose a scheme $U'$ and a surjective smooth morphism $U' \to \mathcal{X}'$. Observe that $U' \to T'$ is smooth and that the projection $T \times _{\mathcal{X}'} U' \to T$ is surjective smooth. Choose an affine scheme $W$ and an étale morphism $W \to T \times _{\mathcal{X}'} U'$ such that $W \to T$ is surjective. Then $W \to T$ is a smooth morphism of affine schemes. After replacing $W$ by a disjoint union of principal affine opens, we may assume there exists a smooth morphism of affines $W' \to T'$ such that $W = T \times _{T'} W'$, see Algebra, Lemma 10.137.20. By More on Morphisms of Spaces, Lemma 76.19.6 we can find a morphism $W' \to U'$ over $T'$ lifting the given morphism $W \to U'$. This finishes the proof. $\square$