Lemma 101.6.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then

$\Delta _{f, 1}$ separated $\Leftrightarrow $ $\Delta _{f, 2}$ closed immersion $\Leftrightarrow $ $\Delta _{f, 2}$ proper $\Leftrightarrow $ $\Delta _{f, 2}$ universally closed,

$\Delta _{f, 1}$ quasi-separated $\Leftrightarrow $ $\Delta _{f, 2}$ finite type $\Leftrightarrow $ $\Delta _{f, 2}$ quasi-compact, and

$\Delta _{f, 1}$ locally separated $\Leftrightarrow $ $\Delta _{f, 2}$ immersion.

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