In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself.
However, I'm having a hard time actually making this explicit, i.e how exactly to move from "it is the universal map to $B$ such that the composition with $f$ is equipped with a nullhomotopy", to "$E$ is the universal space over $B$ equipped with a trivialization of the pullback of the principal bundle to $E$"?
Where to use homotopy invariance?