additive functor induces functor on chain complexes
1. Proposition
Let \(\mathcal{A}, \mathcal{B}\) be abelian categories and \(\mathrm{Ch_*}(\mathcal{A}), \mathrm{Ch_*}(\mathcal{B})\) be the categories of chain complexes Suppose \(\mathcal{F}: \mathcal{A} \rightarrow \mathcal{B}\) is an additive functor, then \(\mathcal{F}\) induces a functor
\begin{align*} \mathcal{F}^*: \mathrm{Ch_*}(\mathcal{A}) \rightarrow \mathrm{Ch_*}(\mathcal{B}) \end{align*}