Lemma 15.86.11. Let $A$ be a ring. Let $E \to D \to F \to E[1]$ be a distinguished triangle of $D(\mathbf{N}, A)$. Let $(E_ n)$, resp. $(D_ n)$, resp. $(F_ n)$ be the system of objects of $D(A)$ associated to $E$, resp. $D$, resp. $F$. Then for every $K \in D(A)$ there is a canonical distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A D_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A F_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n)[1] \]

in $D(A)$ with notation as in Remark 15.86.10.

**Proof.**
This is clear from the construction in Remark 15.86.10 and the fact that $\Delta : D(A) \to D(\mathbf{N}, A)$, $- \otimes ^\mathbf {L} -$, and $R\mathop{\mathrm{lim}}\nolimits $ are exact functors of triangulated categories.
$\square$

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