Lemma 100.48.2. A quasi-separated algebraic stack $\mathcal{X}$ is decent. More generally, if $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is quasi-compact, then $\mathcal{X}$ is decent.

**Proof.**
Namely, if $\mathcal{X}$ is quasi-separated, then any morphism $f : T \to \mathcal{X}$ whose source is a quasi-compact scheme $T$, is quasi-compact, see Lemma 100.7.7. If $\Delta $ is on known to be quasi-compact, then one uses the description

to prove this. Details omitted. $\square$

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