$\require{AMScd}$ On page $412$, Hatcher defines a fibration $F\xrightarrow{}E\xrightarrow{}B$ is called principal if there is a commutative diagram
\begin{CD} F@>{}>>E@>{}>>B\\ @VVV @VVV @VVV\\ \Omega B'@>{}>>F'@>{}>>E'@>{}>>B' \end{CD} where the second row is a fibration sequence and the vertical maps are weak homotopy equivalences. I assume Hatcher adopted this definition as it better served the discussion of Postinkov Towers. However, I have also seen an alternative definition of "Principal" fibration, which seemed more prevalant: a fibration $F\xrightarrow{} E \xrightarrow{} B$ is principal if it is the pullback fibration induced by the path-loop fibration of a "classifying space" $A$, and a "classifying map" $k:B\xrightarrow{ } A$.
It is not clear to me why the two definition are indeed equivalent? Any help on this is greatly appreciated!