Let $G$ be a finite discrete group. A $G$-cover on $X$ is a covering space $p: E \to X$ with group of deck transformations $G$ acting transitively on each fiber (so a principal $G$-bundle).
In Qiaochu Yuan's writeup it's stated that such a cover corresponds to a functor $$\Pi_1(X) \to BG$$ from the fundamental groupoid of $X$ to $BG$, the category with an object $\bullet$ and morphisms $\bullet\to\bullet$ given by elements of $g$.
What does this functor signify geometrically? I can imagine it assigns to a closed loop at $x$ the monodromy at $x$, but given a path $l$ from $x$ to $y$ in $X$, what significance does the element assigned to $l$ have?