There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles.
Thinking of a bundle as a comb with the "teeth" as its fibers and the base as the base space, I am looking for some geometric intuition that would make the theorem intuitively obvious, or at least expectable.
What is the geometric intuition behind the homotopy invariance of fiber bundles?