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, for an appropriate object $G$, 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?Can one give some nontrivial examples in algebraic topology?