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 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?

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