I guess you want something more explicit than saying that the free abelian group on a set is what one gets by applying the left adjoint of the forgetful functor from abelian groups to sets?

Also I am not completely sure what you mean by your last comment - if you are trying to understand the relationship between homology and cohomology via the chain/cochain complexes you might be after the following?

<b>Theorem</b> Let $R$ be a ring and let $P$ be a chain complex of projective $R$-modules such that each $d(P_i)$ is also projective. Then for every $i$ and every $R$-module $N$, there is a non-canonically split short exact sequence
$$ 0 \to Ext^1_R(H_{i-1}(P),N) \to H^i(Hom_R(P,N)) \to Hom_R(H_i(P),N) \to 0$$

So in particular for complexes of free abelian groups one can apply this and it tells you how taking duals alters the cohomology.