The Stacks project

Proof. Since $f$ is a universal homeomorphism, the diagonal $\Delta : X \to X \times _ Y X$ is a surjective closed immersion by Morphisms, Lemmas 29.45.4 and 29.10.2. Since $\Delta $ is also flat, we see that $\Delta $ must be an isomorphism by Morphisms, Lemma 29.26.1. In other words, $f$ is a monomorphism (Schemes, Lemma 26.23.2). Since $f$ is a universal homeomorphism it is certainly quasi-compact. Hence by Descent, Lemma 35.25.1 we find that $f$ is an isomorphism. $\square$