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