Tom wrote "In general such a map E→E′ is not itself a covering space, although this is true in the good case, i.e. when X is connected and locally path-connected and semi-locally simply connected"
I guess, even $X$ is a "deloopable" space (i.e. semilocally simply connected ect.) such a maps $E \to E'$ are not in general covering space. F. ex. a lifting of the universal cover is a morphism of the category being said about (because the identity §X \to X§ is a covering space). It can however never be a covering space. I guess even, "covering" (being a reflexive transitive relation) can never be symmetric, i.e. two non-homeomorphic spaces $X$, $Y$ can never cover each other. Two (non-isomorphic) fields can never be Galois extension of each other