I am currently trying to understand how reductions of structure groups of principal bundles correspond to lifts of classifying maps.
Some definitions:
Let $\rho: H \rightarrow G$ be a homomorphism of compact Lie Groups. A reduction of structure group of a $G$-bundle $P \rightarrow M$ is a pair $(Q, \theta)$, where $Q$ is an $H$-bundle $Q \rightarrow M$, together with an isomorphism of principal $G$-bundles $\theta: Q \times_H G \rightarrow P$. An isomorphism of reductions $(Q,\theta) \rightarrow (Q',\theta')$ is an isomorphism $\psi: Q \rightarrow Q'$ of $H$-bundles, s.t. the induced isomorphism $\psi_G : Q \times_H G \rightarrow Q' \times_H G$ fulfills $\theta = \theta' \circ \psi_G$ (at least this is the definition that makes most sense to me. In the lecture notes Bordism: old and new by Dan Freed making this definition is an exercise).
I am trying to understand the following:
Let $f: M \rightarrow BG$ be a classifying map for $P \rightarrow M$. Then isomorphism classes of reductions are in 1:1 correspondence with homotopy classes of lifts $g:M \rightarrow BH$, where $B\rho \circ g = f$ and homotopy of lifts are homotopies over BG (see again the lecture notes Bordism: old and new for more).
What I do understand is, how one gets a reduction from a lift and the other way around.
But I don't understand on how we get homotopies from isomorphisms. I see how one constructs a homotopy, between two classifying maps of isomorphic bundles as in Notes on Principal Bundles and classifying Spaces, but I don't get, how the fact that they are isomorphisms of reductions gives the possibility to strengthen this to a homotopy of lifts.
Fibrations and classifying spaces are still fairly new to me, so there might be a lot of errors in my attempts above. Please correct me, if anything is wrong and I would really appreciate any sources that could help me improve my understanding of this subject.