I read two versions of discussions on universal bundles. I could not really see how the two definitions are really the same.
From Wiki. The universal bundle in the theory of fiber bundles with structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map M $\to$ BG.
From ncatlab For G a topological group there is a notion of G-principal bundles P→X over any topological space X. Under continuous maps f:X→Y there is a notion of pullback of principal bundles f∗:GBund(Y)→GBund(X). A universal G-principal bundle is a G-principal bundle, which is usually written EG→BG, such that for every CW-complex X the map [X,BG]→GBund(X)/∼ from homotopy classes of continuous functions X→BG given by [f]↦f∗EG, is an isomorphism. In this case one calls BG a classifying space for G-principal bundles. The universal principal bundle is characterized, up to equivalence, by its total space EG being contractible.
Can one explain how the twos are the same? How do we intuitively define Universal Bundle?