Lemma 18.44.2. In the situation above the map to the sheafification

has the following universal property: for any homomorphism of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ such that each local section of $\mathcal{S}$ acts invertibly on $\mathcal{G}$ there exists a unique factorization $(\mathcal{S}^{-1}\mathcal{F})^\# \to \mathcal{G}$. Moreover we have

as sheaves of $(\mathcal{S}^{-1}\mathcal{O})^\# $-modules.

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