@Balarka Ok, I believe the thing I wanted earlier isn't quite, but almost true. First of all, we should work with the category of finite-dim. spaces, because fibers should be finite-dim.. Now, when one has a bundle and looks at a transition maps, they're bundle maps $U\times\mathbb{R}^k\rightarrow U\times\mathbb{R}^k$ that are the identity on the base. This is equivalent to the data of a map $U\rightarrow GL(k)$. The only natural way to turn this into a trivialization of the fiber-wise functored-up bundle is by applying the functor fiber-wise.