I'm reading this paper and I am trying to wrap my head around the bijection found on the bottom of page 2, namely
$$\operatorname{Hom}(\pi_1 (X),G)/G\xrightarrow{\sim} \operatorname{Prin}_G(X)$$
Where $G$ is a discrete group. This is how I see it: An element $\alpha\in \operatorname{Hom}(\pi_1 (X),G)$ is sent to a fiber in $ \operatorname{Prin}_G(X)$. Since $G$ acts freely, each $\alpha$ represents an automorphism of $G$. Modding out by $G$ indicates we should regard each of these fibers (or automorphisms of $G$) in $ \operatorname{Prin}_G(X)$ as the same. How far off am I in my understanding?