If two total spaces are locally homeomorphic to their base spaces, when will a fibre preserving map between them be continuous?

Question

Let $fp = p'g$, where $f$ is continuous, $p$ and $p'$ are local homeomorphisms.

When will $g$ be continuous ?

My reading says it is iff any local (continuous) section $s$ of $p$ over $U$ which is open, then $gs$ is a section of $p'$ over $f(U).$

However, I can't prove it, even can't understand why $gs$ is a section over $f(U).$