Let $X, Y, Z$ be topological spaces with $X$ connected, $p: Y \to X$ and $q: Z \to X$ be covering maps and $f: Y \to Z$ a continuous function, such that $p = q \circ f$. Is $f$ necessarily a covering?
I know that if $X$ is locally connected then it must be, but I am wondering if I can drop that assumption.
What if we don't assume that $X$ is locally connected, but we assume that covering maps $p$ and $q$ have finite fibers? Is $f$ a covering map itself then?