Let A denote an abelian category, Ch(A) denote the corresponding category of chain complex. Then let HoCh(A) denote the category whose objects are the same of Ch(A), but the map between objects are equivalent class that identify maps that are homotopic. The result is Ch(A) is an abelian category but HoCh(A) is no longer abelian.
I even do not know how to start? Which condition of abelian category will fail in this case? Maybe some map in the category does not have kernel?