Lemma 18.44.1. In the situation above the map to the sheafification
\[ \mathcal{O} \longrightarrow (\mathcal{S}^{-1}\mathcal{O})^\# \]
is a homomorphism of sheaves of rings with the following universal property: for any homomorphism of sheaves of rings $\mathcal{O} \to \mathcal{A}$ such that each local section of $\mathcal{S}$ maps to an invertible section of $\mathcal{A}$ there exists a unique factorization $(\mathcal{S}^{-1}\mathcal{O})^\# \to \mathcal{A}$.
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