Lemma 15.73.6. Let $R$ be a ring. Given complexes $K, L$ in $D(R)$ there is a canonical morphism

in $D(R)$ functorial in both $K$ and $L$.

Lemma 15.73.6. Let $R$ be a ring. Given complexes $K, L$ in $D(R)$ there is a canonical morphism

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

in $D(R)$ functorial in both $K$ and $L$.

**Proof.**
This is a special case of Lemma 15.73.5 but we will also prove it directly. Choose a K-flat complex $K^\bullet $ representing $K$ and any complex $L^\bullet $ representing $L$. Choose a quasi-isomorphism $\text{Tot}(K^\bullet \otimes _ R L^\bullet ) \to J^\bullet $ where $J^\bullet $ is K-injective. Then we use the map

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

where the first map is the map from Lemma 15.71.5. $\square$

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