(Comment to [this answer](https://mathoverflow.net/a/111322/))

 Tom Goodwillie wrote "In general such a map $E\to E'$ 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 etc.) such maps $E \to E'$ are not in general covering space. For example 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