The Stacks project

Lemma 85.12.5. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. The category of cartesian $\mathcal{O}$-modules is equivalent to the category of pairs $(\mathcal{F}, \alpha )$ where $\mathcal{F}$ is a $\mathcal{O}_0$-module and

\[ \alpha : (f_{\delta _1^1})^*\mathcal{F} \longrightarrow (f_{\delta _0^1})^*\mathcal{F} \]

is an isomorphism of $\mathcal{O}_1$-modules such that $(f_{\delta ^2_1})^*\alpha = (f_{\delta ^2_0})^*\alpha \circ (f_{\delta ^2_2})^*\alpha $ as $\mathcal{O}_2$-module maps.

Proof. The proof is identical to the proof of Lemma 85.12.4 with pullback of sheaves of abelian groups replaced by pullback of modules. $\square$

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