I feel like this should already have been asked on SE, but I cannot seem to find anything on the matter.
It is well-known that for well-behaved spaces $X$, e.g. paracompact spaces, one can construct for all topological groups $G$ a classifying space $BG$ such that the principal $G$-bundles over $X$ are in bijection with the homotopy classes of maps $[X,BG]$.
This is all well formulated in the toplogical category. However, everywhere I read it seems like people also apply this to smooth principal bundles, i.e. bundles where the base space is a manifold, the group is a Lie group and all maps are smooth.
I cannot seem to understand why this classification would carry over to the smooth category. Is there a simple proof? Or is this classification not valid and did I miss something?