Additive indexing category for chain complexes ($\mathbf{Z}$)

Let $\mathbf{Z}$ have the integers as objects, with $\mathbf{Z}(n,p)=\mathbb{Z}$ for $p=n$ or $p=n-1$ and zero otherwise. Composition is bilinear, the generator of $\mathbf{Z}(n,n)$ is the identity, and the composite of two degree-lowering generators is zero. Additive functors $\mathbf{Z}\to \mathcal{A}$ are exactly chain complexes in the additive category $\mathcal{A}$.