Lemma 70.6.5. Notation and assumptions as in Situation 70.6.1. If
$f$ is universally injective,
$f_0$ is locally of finite type,
then $f_ i$ is universally injective for some $i \geq 0$.
Proof. Recall that a morphism $X \to Y$ is universally injective if and only if the diagonal $X \to X \times _ Y X$ is surjective (Morphisms of Spaces, Definition 67.19.3 and Lemma 67.19.2). Observe that $X_0 \to X_0 \times _{Y_0} X_0$ is of locally of finite presentation (Morphisms of Spaces, Lemma 67.28.10). Hence the lemma follows from Lemma 70.6.4 by considering the morphism $X_0 \to X_0 \times _{Y_0} X_0$. $\square$
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