n-th component functor of chain complexes

Let \(\mathcal{A}\) be pointed category and \(\mathrm{Ch}(\mathcal{A})\) the category of chain complexes. Then for \(n \in \mathbb{N}\), the \(n\)-th component functor is defined as functor

\begin{align*} \mathrm{n-comp}: \mathrm{Ch_*}(\mathcal{A}) \rightarrow& \mathcal{A} \\ (H_n) \mapsto& H_n (f: (H_n) \rightarrow (G_n)) \mapsto& f_n: H_n \rightarrow G_n \end{align*}