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
\xymatrix{ \mathcal{F} \ar[r] \ar[d] & \mathcal{F}^{+} \ar[r] \ar[d] & \mathcal{F}^{++} \ar[d] \\ \mathcal{G} \ar[r] & \mathcal{G}^{+} \ar[r] & \mathcal{G}^{++} }
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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