The Stacks project

Remark 61.10.4. Let $S$ be a scheme contained in a big site $\mathit{Sch}_ h$. Let $F$ be a sheaf of sets on $(\mathit{Sch}/S)_ h$ such that $F(T) = \mathop{\mathrm{colim}}\nolimits F(T_ i)$ whenever $T = \mathop{\mathrm{lim}}\nolimits T_ i$ is a directed limit of affine schemes in $(\mathit{Sch}/S)_ h$. In this situation $F$ extends uniquely to a contravariant functor $F'$ on the category of all schemes over $S$ such that (a) $F'$ satisfies the sheaf property for the h topology and (b) $F'$ is limit preserving. See More on Flatness, Lemma 38.35.4. In this situation Lemma 61.10.3 tells us that $F'$ satisfies the sheaf property for the V topology.


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