Lemma 10.13.3. Let $R \to S$ be a ring map. The functors $\text{Mod}_ S \to \text{Mod}_ R$, $N \mapsto N_ R$ (restriction) and $\text{Mod}_ R \to \text{Mod}_ S$, $M \mapsto M \otimes _ R S$ (base change) are adjoint functors. In a formula

**Proof.**
If $\alpha : M \to N_ R$ is an $R$-module map, then we define $\alpha ' : M \otimes _ R S \to N$ by the rule $\alpha '(m \otimes s) = s\alpha (m)$. If $\beta : M \otimes _ R S \to N$ is an $S$-module map, we define $\beta ' : M \to N_ R$ by the rule $\beta '(m) = \beta (m \otimes 1)$. We omit the verification that these constructions are mutually inverse.
$\square$

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