Let $p:E\rightarrow B$ be a covering map (in particular $p$ is a fiber bundle with discrete fiber). We want to prove the following: Given a commuting diagram of the following form:
$\{0\}\rightarrow E$
$\downarrow\ \ \ \ \ \ \ \ \downarrow$
$I\ \ \rightarrow\ B$
Then there is a unique map $h:I\rightarrow E$, such that the upper and lower triangle commute.
What i have done so far: The existence of the map $h:I\rightarrow E$ is not the problem, since $p$ is a fiber bundle, hence a Serre fibration, hence we have the left lifting property with respect to all inclusions $I^n\times\{0\}\rightarrow I^n\times I$ for all $n\geq0$ by chosing $n=0$ we get our map $h:I\rightarrow E$. For the unqiueness i have problems. Suppose also $k:I\rightarrow E$ does the job. We want to prove that $h(t)=k(t)$ for all $t\in I$. But I only know that $k$ is such a digonal filler. Can someone help me? Thanks a lot.