Lemma 84.12.6. In Situation 84.3.3.

The full subcategory of cartesian abelian sheaves forms a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{total})$. Colimits of systems of cartesian abelian sheaves are cartesian.

Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that the morphisms

\[ f_{\delta ^ n_ j} : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1}) \]are flat. The full subcategory of cartesian $\mathcal{O}$-modules forms a weak Serre subcategory of $\textit{Mod}(\mathcal{O})$. Colimits of systems of cartesian $\mathcal{O}$-modules are cartesian.

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