Lemma 15.68.3. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism

in $D(R)$ functorial in $K, L, M$.

Lemma 15.68.3. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(K, L), M) \]

in $D(R)$ functorial in $K, L, M$.

**Proof.**
Choose a K-injective complex $I^\bullet $ representing $M$, a K-injective complex $J^\bullet $ representing $L$, and a K-flat complex $K^\bullet $ representing $K$. The map is defined using the map

\[ \text{Tot}(\mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet ) \otimes _ R K^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet ), I^\bullet ) \]

of Lemma 15.67.3. We omit the proof that this is functorial in all three objects of $D(R)$. $\square$

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