Lemma 10.14.5. Let $R \to S$ be a ring map. Given $S$-modules $M, N$ and an $R$-module $P$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _ R(M \otimes _ S N, P) = \mathop{\mathrm{Hom}}\nolimits _ S(M, \mathop{\mathrm{Hom}}\nolimits _ R(N, P)) \]

Lemma 10.14.5. Let $R \to S$ be a ring map. Given $S$-modules $M, N$ and an $R$-module $P$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _ R(M \otimes _ S N, P) = \mathop{\mathrm{Hom}}\nolimits _ S(M, \mathop{\mathrm{Hom}}\nolimits _ R(N, P)) \]

**Proof.**
This can be proved directly, but it is also a consequence of Lemmas 10.14.4 and 10.12.8. Namely, we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ R(M \otimes _ S N, P) & = \mathop{\mathrm{Hom}}\nolimits _ S(M \otimes _ S N, \mathop{\mathrm{Hom}}\nolimits _ R(S, P)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ S(M, \mathop{\mathrm{Hom}}\nolimits _ S(N, \mathop{\mathrm{Hom}}\nolimits _ R(S, P))) \\ & = \mathop{\mathrm{Hom}}\nolimits _ S(M, \mathop{\mathrm{Hom}}\nolimits _ R(N, P)) \end{align*}

as desired. $\square$

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