Let $X$, $Y$ and $Z$ be topological spaces and let $f:X \to Z$ and $g:Y \to Z$ be continuous. Let $P$ be the pullback of $X\stackrel{f}{\to}Z\stackrel{g}{\leftarrow} Y$. Let $E$ be another space and let $p:E\to X$ and $q:E \to Y$ be continuous, such that $f \circ p = g \circ q$. By the pullback property, they induce a continuous map $h: E \to P$.
My question: If $f$, $g$, $p$ and $q$ are Hurewicz fibrations, can I conclude that $h$ is again a Hurewicz fibration?