Lemma 75.16.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

The morphism $f$ is étale, and

the morphism $f$ is locally of finite presentation and formally étale.

Lemma 75.16.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

The morphism $f$ is étale, and

the morphism $f$ is locally of finite presentation and formally étale.

**Proof.**
Follows from the case of schemes, see More on Morphisms, Lemma 37.8.9 and étale localization, see Lemma 75.16.2 and Morphisms of Spaces, Lemmas 66.28.4 and 66.39.2.
$\square$

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