# How to lift maps going into the base of a fiber bundle?

Let $p:E\to B$ be a fiber bundle and $f:B'\to B$ a map. Under what conditions does a lift $f':B'\to E$ exist? In the context of covering spaces, I remember a necessary and sufficient condition is that

$f_{\#}(\pi_1(B'))\subseteq p_{\#}(\pi_1(E))$

where $p_{\#}:\pi_1(E)\to \pi_1(B)$ and $f_{\#}:\pi_1(B')\to \pi_1(B)$ are the induced morphisms at the level of the fundamental groups. I am guessing that since covering spaces are a particular case of fiber bundles then at least this condition has to be satisfied, but is it also enough?

Define the pullback of the fiber bundle as $f^*E = \{(x,e) \in B' \times E \mid f(x)=p(e)\}$. This is a fiber bundle over $B'$, and a section of this fiber bundle is the same thing as a lift of the map $f$.