Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor?
I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not sure if I've stated the question clear.) Thanks a lot.
By definition a homology operator is a functor. This only refers to the very basics of definitions in category theory. All it says is that if $id$ is an identity morphism in $cKom$ (whatever $cKom$ may be in your case), then $H_k(id)$ is an identity morhpism in $Ab$, and that if $f:X\to Y $ and $g:Y\to Z$ are morphisms in $cKom$, then $H_k(g\circ f)=H_k(f)\circ H_k(g)$. You prove these identities based on the particular homology construction you have at hand.