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 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.


  [1]: https://i.sstatic.net/LVcQXEdr.png
  [2]: https://i.sstatic.net/DdVZD304.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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