Lemma 15.72.5. Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism

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

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

Proof. Choose a K-flat complex $K^\bullet$ representing $K$, and a K-injective complex $I^\bullet$ representing $L$, and choose any complex $M^\bullet$ representing $M$. Choose a quasi-isomorphism $\text{Tot}(K^\bullet \otimes _ R I^\bullet ) \to J^\bullet$ where $J^\bullet$ is K-injective. Then we use the map

$\text{Tot}\left( K^\bullet \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , I^\bullet ) \right) \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , \text{Tot}(K^\bullet \otimes _ R I^\bullet )) \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , J^\bullet )$

where the first map is the map from Lemma 15.70.3. $\square$

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