The defonition of cohomology of a complex is based on the following:

We have a cpmplex (of appropriate objects) $$0\leftarrow C_0\leftarrow C_1\leftarrow C_2\ldots \leftarrow C_n\ldots$$. Then for an abelian group $G$, we consider the dual complex $$0\rightarrow Hom(C_0,G)\rightarrow Hom(C_1,G)\rightarrow\ldots Hom(C_n,G)\ldots$$. This complex generate the cohomology asdociated to the initoal complex $C_n$.

Now lets consider the following complex:

$$0\leftarrow Hom(G,C_0)\leftarrow Hom(G,C_1)\ldots \leftarrow Hom(G,C_n)\ldots$$


>Is there a terminology for this construction? Is there an analogue to the universal coefficient theorem on order to gove a  relation between the homology of the later complex to the homology of the initial complex for $C_n$? Are there some  application of this construction which shows that the later homology is more useful or more convenient than the initial one?