The Stacks project

Lemma 6.20.1. Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf $\mathcal{O}$-modules. Let $\mathcal{O}^\# $ be the sheafification of $\mathcal{O}$. Let $\mathcal{F}^\# $ be the sheafification of $\mathcal{F}$ as a presheaf of abelian groups. There exists a map of sheaves of sets

\[ \mathcal{O}^\# \times \mathcal{F}^\# \longrightarrow \mathcal{F}^\# \]

which makes the diagram

\[ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{O}^\# \times \mathcal{F}^\# \ar[r] & \mathcal{F}^\# } \]

commute and which makes $\mathcal{F}^\# $ into a sheaf of $\mathcal{O}^\# $-modules. In addition, if $\mathcal{G}$ is a sheaf of $\mathcal{O}^\# $-modules, then any morphism of presheaves of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ (into the restriction of $\mathcal{G}$ to a $\mathcal{O}$-module) factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ where $\mathcal{F}^\# \to \mathcal{G}$ is a morphism of $\mathcal{O}^\# $-modules.

Proof. Omitted. $\square$

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