Reduction of structure group and classifying spaces

Question

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.

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

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

(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,

... what other arrows need I add to the above diagram to get the complete story?

As per Baylee's answer below, it seems convincing to organise the data as follows.

Edit: I am looking to understand whether 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 intuition is that these two data encode equivalent information (but I may be not be exactly right, inasmuch as a proposition of mutual existence is not a 1:1 correspondence).

The question remains how to read the above diagram to infer whether a (lift up to) homotopy $h$ is the same as an isomorphism $\theta_h$.

If further there are any references that explicate the equivalence between these two definitions, please do post them here.

Note: A reminiscent result shows up in Proposition 9.38 of Fre12, but this result remarks on (strict) lifts of $M \to BG$ (taken up to homotopy) rather than its lifts up to homotopy, which are different. I believe only the latter is relevant for the homotopy-theoretic definition I'd like to work with.

Answer 1

To begin, I should mention that the proof of this equivalence is convincingly sketched in Stephen A. Mitchell's "Notes on principal bundles and classifying spaces", see Theorem 10.1 on page 18.

However, the piece you're missing is left as an exercise, so I will elaborate below.

Suppose that there is a lift up to homotopy $\widetilde{P} \colon M \to BH$ of $P \colon M \to BG$. That is, $B\phi \circ \widetilde{P} \simeq P$. We claim that this yields an isomorphism $P \cong Q\times_H G$ for some principal $H$-bundle $Q$.

First, since $\widetilde{P} \colon M \to BH$ exists, we can take $Q$ to be $\widetilde{P}^*EH$. In other words, $Q$ fits in the pullback diagram:

Importantly, $B\phi \colon BH \to BG$ is the classifying map for the principal $G$-bundle $EH\times_HG \to BH$. That is, we have the following pullback square.

Finally, applying the functor $(-)\times_HG$ to the map $p_2$, we obtain the following commutative diagram:

The claim now follows from the universal mapping property of the pullback and the fact than any morphism of principal $G$-bundles is an isomorphism.

Finally, the reverse implication (i.e. that an isomorphism $P \cong Q \times_H G$ gives rise to a homotopy lift of the classifying map of $P$ to $BH$) is clearly explained in Mitchell's notes for admissible $H \subseteq G$.