Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a section in the sense that $p \circ \tilde{f}=\Phi$ where $\Phi$ is an attaching map for a cell $e$ in $X$. Suppose that $c \in [\partial(D^n \times I),F]$ be a given class.
How to show that it is possible to extend $\tilde{f}$ on $\partial(D^n \times I)$ such that the obstruction cocycle associated with this extended $\tilde{f}$ evaluated on $e$ is exactly $c$?
One can find a proof of the similar statement in the context of obstructions for extending maps, see page 176 of these notes (in this context the values of the obstruction cocycle are in $Y$, in our notation it would be $E$-but in the context of fibratons we want these values to be in $F$ thus my question, despite smiliarity, is not a special case of this theory).