The Stacks project

Lemma 85.5.4. Let $X_\lambda , \lambda \in \Lambda $ and $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ be as in Definition 85.5.1. If $Y$ is a quasi-compact algebraic space over $S$, then any morphism $Y \to X$ factors through an $X_\lambda $.

Proof. Choose an affine scheme $V$ and a surjective ├ętale morphism $V \to Y$. The composition $V \to Y \to X$ factors through $X_\lambda $ for some $\lambda $ by the discussion following Definition 85.5.1. Since $V \to Y$ is a surjection of sheaves, we conclude. $\square$

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