Lemma 18.14.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. The category $\textit{Mod}(\mathcal{O})$ is an abelian category. The forgetful functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ is exact, hence kernels, cokernels and exactness of $\mathcal{O}$-modules, correspond to the corresponding notions for abelian sheaves.

Proof. Above we have seen that $\textit{Mod}(\mathcal{O})$ is an additive category, with kernels and cokernels and that $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ preserves kernels and cokernels. By Homology, Definition 12.5.1 we have to show that image and coimage agree. This is clear because it is true in $\textit{Ab}(\mathcal{C})$. The lemma follows. $\square$

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