Lemma 100.6.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then

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

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

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

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