The Stacks project

Proposition 19.8.5. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. The category $\textit{PMod}(\mathcal{O})$ of presheaves of $\mathcal{O}$-modules has functorial injective embeddings.

Proof. We could prove this along the lines of the discussion in Section 19.6. But instead we argue using the theorem above. Endow $\mathcal{C}$ with the structure of a site by letting the set of coverings of an object $U$ consist of all singletons $\{ f : V \to U\} $ where $f$ is an isomorphism. We omit the verification that this defines a site. A sheaf for this topology is the same as a presheaf (proof omitted). Hence the theorem applies. $\square$

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