Let $H, G$ be topological groups and $\phi : H \to G$ a group homomorphism. Let $M$ be a paracompact topological space. For any principal $G$-bundle $P \to M$, a reduction (or sometimes 'lift') of its structure group to $H$ is a principal $H$-bundle $Q \to M$ and a ($G$-equivariant) isomorphism of fibre bundles $Q \times_{\phi} G\cong P$, i.e. [![reductions of structure group via bundles][1]][1] The homotopy-theoretic definition of reductions is as follows. Let $BG$ be the classifying space for principal $G$-bundles. Let $B\phi : BH \to BG$ be a map of classifying spaces corresponding to $\phi$. Then given a classifying map $\tau_G : M \to BG$, a ($H$-)reduction of the structure group (of the principal $G$-bundle $\tau_G$ classifies, call it $P$) is a lift up to homotopy of this map to $BH$, i.e. a map $\tau_H : M \to BH$ and a homotopy $h : B\phi \circ \tau_H \to \tau_G$, so that we have the diagram [![via classifying spaces][2]][2] I'd like to see how the homotopy-theoretic definition is equivalent to the principal bundle definition. For example, if I spell out the principal $G$-bundle $P \to M$ here, I presumably get a diagram of the sort [![P added][3]][3] (where I'm not sure if the existence of $P \to (B\phi)^*EG$ follows directly from the admission of an $H$-reduction or not) Then the (first) question would be where the principal $H$-bundle $Q \to M$ fits into the above. If I include the pertinent pullback square in blue, [![Q added][4]][4] ... what other arrows need I add to the above [diagram][5] to get the complete story? If further there are any references that explicate the equivalence between these two definitions, please do post them here. Edit: I am looking to show that each homotopy $h : B\phi \circ \tau_H \to \tau_G$ leads to an isomorphism of principal $G$-bundles $\theta_h : Q \times_{\phi} G \to P$ and vice-versa. My understanding is that these two data encode equivalent information. As per Baylee's answer below, they should be organised as follows. [![full diagram][6]][6] The question remains how to read the above diagram to infer that a (lift up to) homotopy $h$ is the same as an isomorphism $\theta_h$. [1]: https://i.sstatic.net/LVcQXEdr.png [2]: https://i.sstatic.net/KnPpjqvG.png [3]: https://i.sstatic.net/7juMZ3eK.png [4]: https://i.sstatic.net/KnhIVKtG.png [5]: https://q.uiver.app/#q=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 [6]: https://i.sstatic.net/UDet9cQE.png