Proof.
The key ingredient of the proof is More on Morphisms, Lemma 37.12.1 which (almost) says that a morphism of schemes of finite type over S satisfying (98.13.2.1) is a smooth morphism. The other arguments of the proof are essentially bookkeeping.
Let V be a scheme over S and let y be an object of \mathcal{Y} over V. Let Z be an algebraic space representing the 2-fibre product \mathcal{Z} = \mathcal{X} \times _{f, \mathcal{X}, y} (\mathit{Sch}/V)_{fppf}. We have to show that the projection morphism Z \to V is smooth, see Algebraic Stacks, Definition 94.10.1. In fact, it suffices to do this when V is an affine scheme locally of finite presentation over S, see Criteria for Representability, Lemma 97.5.6. Then (\mathit{Sch}/V)_{fppf} is limit preserving by Lemma 98.11.3. Hence Z \to S is locally of finite presentation by Lemmas 98.11.2 and 98.11.3. Choose a scheme W and a surjective étale morphism W \to Z. Then W is locally of finite presentation over S.
Since f satisfies (98.13.2.1) we see that so does \mathcal{Z} \to (\mathit{Sch}/V)_{fppf}, see Lemma 98.13.5. Next, we see that (\mathit{Sch}/W)_{fppf} \to \mathcal{Z} satisfies (98.13.2.1) by Lemma 98.13.6. Thus the composition
(\mathit{Sch}/W)_{fppf} \to \mathcal{Z} \to (\mathit{Sch}/V)_{fppf}
satisfies (98.13.2.1) by Lemma 98.13.4. More on Morphisms, Lemma 37.12.1 shows that the composition W \to Z \to V is smooth at every finite type point w_0 of W. Since the smooth locus is open we conclude that W \to V is a smooth morphism of schemes by Morphisms, Lemma 29.16.7. Thus we conclude that Z \to V is a smooth morphism of algebraic spaces by definition.
\square
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