Lemma 100.9.10 (0506)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 100: Properties of Algebraic Stacks
Section 100.9: Immersions of algebraic stacks
Lemma 100.9.10 (cite)

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Lemma 100.9.10. For any immersion $i : \mathcal{Z} \to \mathcal{X}$ there exists a unique locally closed substack $\mathcal{X}' \subset \mathcal{X}$ such that $i$ factors as the composition of an equivalence $i' : \mathcal{Z} \to \mathcal{X}'$ followed by the inclusion morphism $\mathcal{X}' \to \mathcal{X}$. If $i$ is a closed (resp. open) immersion, then $\mathcal{X}'$ is a closed (resp. open) substack of $\mathcal{X}$.

Proof. Omitted. $\square$

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