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

We have a complex (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 generates the cohomology associated to the initial complex $C_n$.

Now let's consider the following complex:

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

Is there any terminology for this construction? Is there an analogue of the universal coefficient theorem in order to give a relation between the homology of the latter complex and the homology of the initial complex for $C_n$? Are there any applications of this construction which show that the latter homology is more useful or more convenient than the initial one?