Remark 84.12.7 (Warning). Lemma 84.12.6 notwithstanding, it can happen that the category of cartesian $\mathcal{O}$-modules is abelian without being a Serre subcategory of $\textit{Mod}(\mathcal{O})$. Namely, suppose that we only know that $f_{\delta _1^1}$ and $f_{\delta _0^1}$ are flat. Then it follows easily from Lemma 84.12.5 that the category of cartesian $\mathcal{O}$-modules is abelian. But if $f_{\delta _0^2}$ is not flat (for example), there is no reason for the inclusion functor from the category of cartesian $\mathcal{O}$-modules to all $\mathcal{O}$-modules to be exact.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).