Lemma 46.7.3. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. The inclusion functor

\[ \textit{Adeq}(\mathcal{O}) \to \textit{Mod}((\mathit{Sch}/U)_\tau , \mathcal{O}) \]

has a right adjoint $A$^{1}. Moreover, the adjunction mapping $A(\mathcal{F}) \to \mathcal{F}$ is an isomorphism for every adequate module $\mathcal{F}$.

**Proof.**
By Topologies, Lemma 34.7.11 (and similarly for the other topologies) we may work with $\mathcal{O}$-modules on $(\textit{Aff}/U)_\tau $. Denote $\mathcal{P}$ the category of module-valued functors on $\textit{Alg}_ A$ and $\mathcal{A}$ the category of adequate functors on $\textit{Alg}_ A$. Denote $i : \mathcal{A} \to \mathcal{P}$ the inclusion functor. Denote $Q : \mathcal{P} \to \mathcal{A}$ the construction of Lemma 46.4.1. We have the commutative diagram

46.7.3.1
\begin{equation} \label{adequate-equation-categories} \vcenter { \xymatrix{ \textit{Adeq}(\mathcal{O}) \ar[r]_-k \ar@{=}[d] & \textit{Mod}((\textit{Aff}/U)_\tau , \mathcal{O}) \ar[r]_-j & \textit{PMod}((\textit{Aff}/U)_\tau , \mathcal{O}) \ar@{=}[d] \\ \mathcal{A} \ar[rr]^-i & & \mathcal{P} } } \end{equation}

The left vertical equality is Lemma 46.5.3 and the right vertical equality was explained in Section 46.3. Define $A(\mathcal{F}) = Q(j(\mathcal{F}))$. Since $j$ is fully faithful it follows immediately that $A$ is a right adjoint of the inclusion functor $k$. Also, since $k$ is fully faithful too, the final assertion follows formally.
$\square$

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