The Stacks project

Remark 13.33.2. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n, f_ n)$ be a system of objects of $\mathcal{D}$. We may think of a derived colimit as an object $K$ of $\mathcal{D}$ endowed with morphisms $i_ n : K_ n \to K$ such that $i_{n + 1} \circ f_ n = i_ n$ and such that there exists a morphism $c : K \to \bigoplus K_ n$ with the property that

\[ \bigoplus K_ n \xrightarrow {1 - f_ n} \bigoplus K_ n \xrightarrow {i_ n} K \xrightarrow {c} \bigoplus K_ n[1] \]

is a distinguished triangle. If $(K', i'_ n, c')$ is a second derived colimit, then there exists an isomorphism $\varphi : K \to K'$ such that $\varphi \circ i_ n = i'_ n$ and $c' \circ \varphi = c$. The existence of $\varphi $ is TR3 and the fact that $\varphi $ is an isomorphism is Lemma 13.4.3.

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