Lemma 101.3.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent

$f$ is locally separated, and

$\Delta _ f$ is an immersion.

Lemma 101.3.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent

$f$ is locally separated, and

$\Delta _ f$ is an immersion.

**Proof.**
The statements “$f$ is locally separated”, and “$\Delta _ f$ is an immersion” refer to the notions defined in Properties of Stacks, Section 100.3. Proof omitted. Hint: Argue as in the proofs of Lemmas 101.3.5 and 101.3.6.
$\square$

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