I know the universal coefficients theorem (homological version) helps us to compare homology (co-homology) over different coefficients ($\mathbb{Z}$-modules), by:
$$ H_n(X;G)\cong \big[ H_n(X) \otimes_\mathbb{Z} G \big] \oplus \text{Tor}^1(H_{n-1}(X),G) $$
But is there a way to connect the chains over different coefficients? I know that $C_n(X;G):=\text{Free}_G\big( X(n) \big)$, and when $G$ is a $\mathbb{Z}$-module, we have that:
$$C_n(X;G)\cong \text{Free}_{\mathbb{Z}}\big( X(n) \big) \otimes_\mathbb{Z} G$$
But can we move from one $\mathbb{Z}$-module to another? More concretely, I have an element $\gamma\in C_{n-1}(X)$ and I take $\gamma' \in C_{n-1}(X;G)$ with appropriate coefficients in $G$. I know that there exists $\omega' \in C_n(X;G)$ such that $\partial(\omega')=\gamma'$. Using this, I want to find $\omega \in C_n(X)$ such that:
$$ \partial(\omega)=\gamma \quad \text{as a } \; G-\text{module morphism} $$
Is there some "diagram chasing" way to obtain such an element?