Let $X$ be a Cw complex basepointed at $x$. Delooping the covering fibration gives the fibration I'm interested $X \to K(\pi_1(X),1)$ with homotopy fiber $\tilde{X}$.
The explicit construction works by turning $X \to K(\pi_1(X),1)$ into a fibration: The total space is the paths in $K(\pi_1(X),1)$ starting somewhere in $X$. The projection map sends a path to its endpoint.
The fiber is the paths in $K(\pi_1(X),1)$ starting somewhere in $X$ and ending at $x \in X$.
We can explicitly see that the constructed fiber is homotopy equivalent to $\tilde{X}$(this viewpoint is useful when describing the monodromy action). Here is the correspondence. Take a path $\gamma$ in the constructed fiber. The path is homotopic relative the start and endpoint to a path lying completely in $X$(b/c $\pi_1(X)=\pi_1(K(\pi_1,1))$). Uniquely lift the path so that the endpoint is lifted to the basepoint of $\tilde{X}$. Record the starting point of the path.
Question: What is the map from $\tilde{X}$ to the fiber?