i attempted the first 100 pages of Walter Poor's Differential Geometric Structures, taking me about two months, and the most difficulty i had was trying to understand the utility(?) of the pullback bundle. its possible that its just a very difficult subject and the only thing that's needed is some time to grasp the ideas, or at least the notation, but its still a very slow subject and i wonder what prerequisites are needed to make it go smoother. it takes a half page just to set up every new idea. he uses the idea in several constructions and several commutation diagram isomorphisms, but i still don't see their full nature. the only intuition i have on the idea is to see it as a kind of 'pre image' of the fibre bundle structure. for example, the structure of a cube might be seen as a pre image of a family of tangent planes along a curve inside a manifold. but why not just do this (a mapping between preimage and image), instead of constructing a commutation diagram and then building catagorical isomorphisms between the diagrams? or am i asking the wrong question? it seems strange that wikipedia describes this stuff way too lightly, as if its such constructions are self evident. thankyou for any help.