I was working in this part of a proposition in Spanier's book.
Let $A,B,E$ topological spaces. If we have two maps $f:A\to B, p:E\to B$ where $p$ is a fibration and $f^*p$ is the fibered product with $\pi_1: f^*p\to A$ and $\pi_2: f^*p\to E$, then $f$ lifts to a map $F:A\to E$ iff $\pi_1$ has a section.
I could do one direction easily: If $\pi_1$ has a section $s$ just define $F=\pi_2 s$.
I had trouble with the second direction: If the lifting $F$ exists, I want to define a map $s:A\to f^*p$. I first thought I could choose $s=iF$ where $i$ is the inclusion $E\to \{a\}\times E$ (where $a$ is a fixed point of $A$) but it's not even well defined, and even if it were, $\pi_1 s$ would be constant. How could I do this part?
Note that $f^*E$ is the pullback of the two maps $f:A\rightarrow B\leftarrow E:p$. So, no matter what space you actually take it to be, it will be homeomorphic to
$$f^*E\cong \{(a,e)\in A\times E\mid f(a)=p(e)\}\subseteq A\times E$$
with the subspace topology. The maps $\pi_1:f^*E\rightarrow A$, $\pi_2:f^*E\rightarrow E$ are given by the restrictions of the projections onto each factor, so are clearly continuous.
If a lifting $F:A\rightarrow E$ of $f$ exists, then define
$$s:A\rightarrow f^*E,\qquad s(a)=(a,F(a)).$$
This map is well defined since
$p(F(a))=f(a),$
by assumption of the lifting property of $F$. Moreover it is continuous since it is the corestriction of the continuous composition
$$s':A\xrightarrow{\Delta} A\times A\xrightarrow{1\times F}A\times E,$$
where $\Delta$ is the diagonal map.
Clearly
$\pi_1\circ s(a)=a,$
so $s$ is indeed a section of $\pi_1:f^*E\rightarrow A$.