colimit of chain complexes

Let \(\mathcal{C}\) be a pointed category and \(\mathrm{Ch_*}(\mathcal{C})\) the category of chain complexes. Suppose there exists a diagram \(J: \mathcal{J} \rightarrow \mathrm{Ch_*}(\mathcal{C})\), then a colimit - if it exists - is determined by each component

\begin{align*} \mathrm{colim}_{J} = (... \rightarrow \mathrm{colim}_{J_n} \rightarrow \mathrm{colim}_{J_n} \rightarrow ... ) \end{align*}

analogous (but not dually) to limit of chain complexes

alternatively:

• colimit of functors for a cocomplete codomain