Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-back of $h$ via $p$.

Then why do $h$ and $\tilde{h}$ have the same fibres? Since $\tilde{X}$ is the covering space of $X$, shouldn't the fiber of $h$ be the quotient of the fiber of $\tilde{h}$?

The statement is from here p.4, last paragraph.