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 étale 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).