The Stacks project

Lemma 6.30.7. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Denote $\mathop{\mathit{Sh}}\nolimits (\mathcal{B})$ the category of sheaves on $\mathcal{B}$. There is an equivalence of categories

\[ \mathop{\mathit{Sh}}\nolimits (X) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) \]

which assigns to a sheaf on $X$ its restriction to the members of $\mathcal{B}$.

Proof. The inverse functor in given in Lemma 6.30.6 above. Checking the obvious functorialities is left to the reader. $\square$

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