Q:

Cohomology from homology with coefficients

The usual presentation of the coboundary map $\delta: \Delta^i(X;G)\to\Delta^{i+1}(X;G)$ is that it is the dual map of the boundary homomorphism $\partial: \Delta_{i+1}(X)\to\Delta_i(X)$. (Hatcher)
In Hatcher, the chain group $\Delta_i(X)$ is taken to be  free abelian with basis the $i$-simplices of $X$.
Can we consider more general $\Delta_i(X)$ with coefficients, say in another abelian group $H$?  How will that change things, and how will the coefficients $H$ "interact" with the coefficients $G$? 
For instance, if $c$ is a $n+1$-chain (with coefficients in $H$), and $\phi$ is a $n+1$-cohain, then is $\delta\phi(c)$ well-defined?  Using the usual formula will give a linear combination of terms in $G$ with coefficients in $H$.
Thanks. 

A:

Note that $\hom_\mathbb Z(\Delta(X)\otimes H,G) = \hom_\mathbb Z(\Delta(X),\hom(H,G))$ by adjunction. 