Lemma 59.51.9. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a directed limit of schemes with affine transition morphisms $f_{i'i}$ and projection morphisms $f_ i : X \to X_ i$. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. Then

1. there are canonical maps $\varphi _{i'i} : f_{i'i}^{-1}f_{i, *}\mathcal{F} \to f_{i', *}\mathcal{F}$ such that $(f_{i, *}\mathcal{F}, \varphi _{i'i})$ is a system of sheaves on $(X_ i, f_{i'i})$ as in Definition 59.51.1, and

2. $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}f_{i, *}\mathcal{F}$.

Proof. Via Topologies, Lemma 34.4.12 and Lemma 59.51.2 this is a special case of Sites, Lemma 7.18.5. $\square$

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