Uniqueness of possibly non-simply connected universal covering spaces. 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.

An important fact I missed is that every universal cover, in the sense of Spanier, is regular. So if $p\colon\tilde{X}\to X$ is a universal cover, then $p_{*}\pi(\tilde{X},\tilde{x}_0)=p_{*}\pi(\tilde{X},\tilde{x}_0')$ provided that $\tilde{x}_0,\tilde{x}_0'$ are in the same fiber of $p$. This immediately shows that there is always a fiber-preserving homeomorphism between two universal covers.
