Let $X, E_1, E_2$ be path-connected topological spaces with coverings $f_1:E_1 \to X$ and $f_2:E_2 \to X$ where we have additionally a continuous function $\phi:E_1 \to E_2$ such that the composition of $f_2$ and $\phi$ is equal to $f_1$. Show that $\phi$ is also a covering.
First we note that $\phi$ is surjective since we can lift paths from $X$ to $E_1$ and $E_2$ respectively and they are unique, and by using in addition the equality $f_1 = f_2 \circ \phi$, the surjectivity follows. But how to continue from here? Any help would be appreciated, thanks.