Homology object ($H_n(A_{\bullet })$)

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.