chain homology functor

Let \(\mathcal{A}\) be an abelian category and \(\mathrm{Ch}(\mathcal{A})\) the category of chain complexes. Then the \(n\)-th chain homology functor is defined as

\begin{align*} H_n: \mathrm{Ch}(\mathcal{A}) \rightarrow& \mathcal{A} \\ V \mapsto& Z_n(V)/B_n(V) = \mathrm{ker}(\partial_{n-1})/\mathrm{im}(\partial_n) (f: V \rightarrow W) \mapsto& (f: Z_n(v)/B_n(V) \rightarrow Z_n(W)/B_n(W)) \end{align*}