Lemma 47.6.3. Let $(R, \mathfrak m, \kappa )$ be an artinian local ring. Let $E$ be an injective hull of $\kappa $. The functor $D(-) = \mathop{\mathrm{Hom}}\nolimits _ R(-, E)$ induces an exact anti-equivalence $\text{Mod}^{fg}_ R \to \text{Mod}^{fg}_ R$ and $D \circ D \cong \text{id}$.
Proof. We have seen that $D \circ D = \text{id}$ on $\text{Mod}^{fg}_ R$ in Lemma 47.6.2. It follows immediately that $D$ is an anti-equivalence. $\square$
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