Complex in an abelian category

A complex in an abelian category is a sequence $\cdots \to A_{n+1}\stackrel{d_{n+1}}{\to }{A}_n\stackrel{d_n}{\to }{A}_{n-1}\to \cdots$ with $d_n{d}_{n+1}=0$.

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 $\mathcal{A}$.

The homology object is $H_n=\ker d_n/\mathrm{im}{d}_{n+1}$, formed categorically as the cokernel of the image-to-kernel monomorphism.

Let $q:C_n\to \mathrm{coker}{d}_{n+1}$. Since $d_n{d}_{n+1}=0$, there is an induced ${\bar{d}}_n:\mathrm{coker}{d}_{n+1}\to C_{n-1}$. In an abelian category,The left side is the usual homology construction and the right side is its construction in the opposite category, so the definition of homology is self-dual.