Complex geometry – Fiber identical to the induced coverage map

Consider a holomorphic map $ h: X to E $ between compact, connected and complex analytic varieties $ p: tilde {E} to E $ to be universal coverage, and denote by $ tilde {h}: tilde {X} to tilde {E} $ the withdrawal of $ h $ via $ p $.

So why do $ h $ and $ tilde {h} $ have the same fibers? Since $ tilde {X} $ is the cover space of $ X $, should not the fiber of $ h $ to be the quotient of the fiber of $ tilde {h} $?

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