I've some trouble understanding this explanation of the proof of lemma 4.7 in Davis and Kirk "Lecture Notes in Algebraic Topology".
Lemma 4.7 Any local coefficient system $A\hookrightarrow E \to B$ is of the form $$ A \hookrightarrow \tilde{B}\times_{\pi_1B} A \to B $$
They build the holodromy map $\pi_1 B \to \text{Aut}(A)$ and then claim that using a standard covering space argument one is done.
I noticed that this problem was addressed in this old question but there is a passage in the answer which are unclear to me:
it's claimed that, after picking $x \in B$, for every choice of $a \in p^{-1}(x)$ there is an associated sub-cover of $E$. How? I mean, the condition given is a fibre-wise condition, so I need to make a choice for every fibre, why all these choices glue together in a sub-cover?
My guess is to somewhat express this as a quotient of $E$ by some subgroup of the deck-transformation group, but it's unclear to me how to proceed.
NB: Since the answer is very old, I decided to ask for help in a separate question.