Let $X$ be a connected and locally path-connected space. In Spanier's book, the universal cover of $X$ is defined to be a connected covering space $p\colon\tilde{X}\to X$ such that for any covering $q\colon Y\to X$ there is a fiber-preserving map $f\colon \tilde{X}\to Y$.
The author stated without proof that any two universal covers are equivalent meaning that there is a fiber-preserving homeomorphism between two universal covering spaces.
Of course, for simply-connected universal covers, this is not a problem. Besides, if the base space is also semi-locally simply-connected, then every universal covering is simply-connected.
My Question is: Is Spanier's statement true, or is it just a mistake in the book?
I've asked a somewhat related question here.