Lemma 67.4.13. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$.
The morphism $f$ is locally separated.
The morphism $f$ is (quasi-)separated in the sense of Definition 67.4.2 above if and only if $f$ is (quasi-)separated in the sense of Section 67.3.
In particular, if $f : X \to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition 67.4.2 if and only if $f$ is (quasi-)separated as a morphism of schemes.
Proof. This is the equivalence of (1) and (2) of Lemma 67.4.12 combined with the fact that any morphism of schemes is locally separated, see Schemes, Lemma 26.21.2. $\square$
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