Lemma 15.89.5. Let $R$ be a ring. Let $K \in D(R)$ be pseudo-coherent. Let $(M_ n)$ be an inverse system of $R$-modules. Then $R\mathop{\mathrm{lim}}\nolimits K \otimes _ R^\mathbf {L} M_ n = K \otimes _ R^\mathbf {L} R\mathop{\mathrm{lim}}\nolimits M_ n$.

**Proof.**
Consider the defining distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits M_ n \to \prod M_ n \to \prod M_ n \to R\mathop{\mathrm{lim}}\nolimits M_ n[1] \]

and apply Lemma 15.62.6. $\square$

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