Lemma 46.4.2. Let $A$ be a ring. 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. Then

$i$ is fully faithful, exact, and its image is a weak Serre subcategory,

$\mathcal{P}$ has enough injectives,

the functor $Q$ is a right adjoint to $i$ hence left exact,

$Q$ transforms injectives into injectives,

$\mathcal{A}$ has enough injectives.

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