Self-duality of homology
= Self-duality of homology
Let $q:C_n\to\operatorname{coker}d_{n+1}$. Since $d_nd_{n+1}=0$, there is an induced $\bar d_n:\operatorname{coker}d_{n+1}\to C_{n-1}$. In an <abelian category>,
$$
\operatorname{coker}\bigl(\operatorname{im}d_{n+1}\to\ker d_n\bigr)
\cong
\ker\bar d_n.
$$
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.