Let $F: Cov_B \to O(\pi_1(B,b))$ be a functor from the category of covering spaces over B to the orbit category of the fundamental group such that $F((q: E \to B)) \to q^{-1}(b)$. I am not sure how to prove that $F$ is fully faithful and essentially surjective.
I was told that $q^{-1}(b) \cong \pi_1(B,b)/q_*(\pi_1(E,e))$ but I do not see why this is true or how it helps.