I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a Postnikov tower of principal fibrations, which is a Postnikov tower

$\require{AMScd}$ \begin{CD} \vdots & & \vdots\\ @| & @VVV \\ X @>>> X_3 \\ @| & @VVV \\ X @>>> X_2\\ @| & @VVV\\ X @>>> X_1 \end{CD} where any map $ X_n \rightarrow X_{n-1}$ is such that exists a fibration sequence $F\rightarrow E\rightarrow B$ and weak homotopy equivalences $X_n\rightarrow F$ and $X_{n-1}\rightarrow E$ such that

$\require{AMScd}$
\begin{CD}
X_n @>>> X_{n-1} \\
@VVV & @VVV \\
F @>>> E@>>> B\\
\end{CD}
commutes.

The problem is that when a few pages later he deals with the problem of finding a map $W\rightarrow X_n$ to complete the commutative diagram
$\require{AMScd}$
\begin{CD}
A @>>> X_n \\
@VVV & @VVV \\
W @>>> X_{n-1}\\
\end{CD}
where $(W,A)$ is a $CW$ pair and $A\rightarrow W$ is the inclusion, he supposes that $X_n$ actually is the pullback of the path fibration $PB\rightarrow B$ via $X_{n-1}\rightarrow B$.
Why can he suppose this?