The Stacks project

Proof. We will use Morphisms, Lemmas 29.25.8 and 29.25.6 without further mention. Write $X \to Y$ as the composition $X \to X \times _ S Y \to Y$. The second morphism is flat as the base change of the flat morphism $X \to S$. The first is the base change of the flat morphism $Y \to Y \times _ S Y$ by the morphism $X \times _ S Y \to Y \times _ S Y$, hence flat. Thus $X \to Y$ is flat. The morphism $X \times _ Y X \to X \times _ S X$ is an immersion. Thus Lemma 37.64.3 implies, that since $X$ is flat over $X \times _ S X$ it follows that $X$ is flat over $X \times _ Y X$. $\square$