Definition 18.9.1. Let $\mathcal{C}$ be a category, and let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$.

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*presheaf of $\mathcal{O}$-modules*is given by an abelian presheaf $\mathcal{F}$ together with a map of presheaves of sets\[ \mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F} \]such that for every object $U$ of $\mathcal{C}$ the map $\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an $\mathcal{O}(U)$-module structure on the abelian group $\mathcal{F}(U)$.

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*morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of $\mathcal{O}$-modules*is a morphism of abelian presheaves $\varphi : \mathcal{F} \to \mathcal{G}$ such that the diagram\[ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi } & \mathcal{F} \ar[d]^{\varphi } \\ \mathcal{O} \times \mathcal{G} \ar[r] & \mathcal{G} } \]commutes.

The set of $\mathcal{O}$-module morphisms as above is denoted $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$.

The category of presheaves of $\mathcal{O}$-modules is denoted $\textit{PMod}(\mathcal{O})$.

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