Consider a continuous map $\alpha:A\to B$ with the homotopy lifting propery and the unique path lifting property. Consider the induced fiber functor $$F:\pi_1B\longrightarrow \mathsf{Set}$$ taking a point $b\in B$ to its fiber $\alpha^{-1}(b)$ and acting on homotopy classes by unique path lifting.
If $A,B$ are locally path connected and moreover $B$ is semilocally simply connected, HLP+UPL are equivalent to $\alpha$ being a covering map. Under these conditions:
Is $F$ being fully faithful by any chance equivalent, or at least related to $\alpha$ being a universal covering?
Fullness means every set function $\alpha^{-1}(b)\to \alpha^{-1}(b^\prime)$ is induced by some homotopy class of path while faithfulness means distinct homotopy classes induce distinct lifting functions.