Lemma 15.59.15. Let $R$ be a ring. Let $K^\bullet , L^\bullet , M^\bullet $ be complexes of $R$-modules. There is a canonical isomorphism

functorial in all three complexes.

Lemma 15.59.15. Let $R$ be a ring. Let $K^\bullet , L^\bullet , M^\bullet $ be complexes of $R$-modules. There is a canonical isomorphism

\[ (K^\bullet \otimes _ R^\mathbf {L} L^\bullet ) \otimes _ R^\mathbf {L} M^\bullet = K^\bullet \otimes _ R^\mathbf {L} (L^\bullet \otimes _ R^\mathbf {L} M^\bullet ) \]

functorial in all three complexes.

**Proof.**
Replace the complexes by K-flat complexes and use the associativity constraint in Section 15.58.
$\square$

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