Lemma 15.86.4. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n)$ be a system of objects of $\mathcal{D}$. Let $K$ be a derived colimit of the system $(K_ n)$. Then for every $L$ in $\mathcal{D}$ we have a short exact sequence

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K_ n, L[-1]) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, L) \to \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K_ n, L) \to 0 \]

**Proof.**
This follows from Derived Categories, Definition 13.33.1 and Lemma 13.4.2, and the description of $\mathop{\mathrm{lim}}\nolimits $ and $R^1\mathop{\mathrm{lim}}\nolimits $ in Lemma 15.86.1 above.
$\square$

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