Lemma 46.5.3. Let $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. The category of adequate $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau $ is equivalent to the category of adequate module-valued functors on $\textit{Alg}_ A$.
Proof. Given an adequate module $\mathcal{F}$ the functor $F_{\mathcal{F}, A}$ is adequate by Lemma 46.5.2. Given an adequate functor $F$ we choose an exact sequence $0 \to F \to \underline{M} \to \underline{N}$ and we consider the $\mathcal{O}$-module $\mathcal{F} = \mathop{\mathrm{Ker}}(M^ a \to N^ a)$ where $M^ a$ denotes the quasi-coherent $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $ associated to the quasi-coherent sheaf $\widetilde{M}$ on $S$. Note that $F = F_{\mathcal{F}, A}$, in particular the module $\mathcal{F}$ is adequate by definition. We omit the proof that the constructions define mutually inverse equivalences of categories. $\square$
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