  For any ring $R$, the Auslander-Gruson-Jensen functor is the exact
contravariant functor
$$\textsf{D}_A:\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})\longrightarrow(\textsf{mod}(R^{op}),\textsf{Ab})$$
sending representable functors $(X,\hspace{0.05cm}\underline{\ \
}\hspace{0.1cm} )$ to tensor functors $X\otimes\hspace{0.05cm}\underline{\ \
}\hspace{0.1cm} $. We show that this functor admits a fully faithful left
adjoint $\textsf{D}_L$ and a fully faithful right adjoint $\textsf{D}_R$. The
left adjoint $$\textsf{D}_L\:(\textsf{mod}(R^{op}),\textsf{Ab})\longrightarrow
\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$$ induces an equivalence of categories
$$\frac{\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})}{\{F\ |\ \textsf{D}_A
F=0\}}\cong(\textsf{mod}(R^{op}),\textsf{Ab})^{op}$$ where $\{F \ |\
\textsf{D}_A F=0\}$ is the Serre subcategory of
$\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$ consisting of all functors $F$
arising from pure exact sequences. As a result, the functor $\textsf{D}_A$ is
seen to be a Serre localization functor. The right adjoint
$$\textsf{D}_R:(\textsf{mod}(R^{op}),\textsf{Ab})\longrightarrow
\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$$ together with $\textsf{D}_A$
restricts to the well known Auslander-Gruson-Jensen duality.
