Proposition 7.10.12. The canonical map $\mathcal{F} \to \mathcal{F}^\# $ has the following universal property: For any map $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf of sets, there is a unique map $\mathcal{F}^\# \to \mathcal{G}$ such that $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ equals the given map.

**Proof.**
By Lemma 7.10.4 we get a commutative diagram

and by Theorem 7.10.10 the lower horizontal maps are isomorphisms. The uniqueness follows from Lemma 7.10.8 which says that every section of $\mathcal{F}^\# $ locally comes from sections of $\mathcal{F}$. $\square$

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