Given a (co/contravariant) functor $F$ from the simplicial category $\Delta$ to an abelian category $A$, we can form its Cech complex (or "alternating face map complex" on the nLab), i.e. $CF^n=F([n])$, and $\partial^n$ is $\sum_{i=0}^n(-1)^iF(\delta^n_i)$, where $\delta^n_i$ is the face map.
By the Dold-Kan correspondence, we can then reconstruct a (co)simplicial object $\Gamma CF$ which has this $CF^\bullet$ as its Moore complex. In particular, its Cech complex is a direct sum of $CF^\bullet$ and a nullhomotopic complex, and is homotopy equivalent to $CF^\bullet$.
So my question is: can we say anything more direct about relationship between our $F$ and $\Gamma CF$ than "they have homotopy-equivalent Cech complexes"? More generally, what relation does homotopy equivalence impose on (co)simplicial objects?
The answer, if anybody is still wondering, is simplicial homotopy.