Lemma 87.9.3. Let $X_\lambda , \lambda \in \Lambda $ and $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ be as in Definition 87.9.1. Then $X_\lambda \to X$ is representable and a thickening.

**Proof.**
The statement makes sense by the discussion in Spaces, Section 65.3 and 65.5. By Lemma 87.9.2 the morphisms $X_\lambda \to X$ are representable. Given $U \to X$ where $U$ is a scheme, then the discussion following Definition 87.9.1 shows that Zariski locally on $U$ the morphism factors through some $X_\mu $ with $\lambda \leq \mu $. In this case $U \times _ X X_\lambda = U \times _{X_\mu } X_\lambda $ so that $U \times _ X X_\lambda \to U$ is a base change of the thickening $X_\lambda \to X_\mu $.
$\square$

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