Let $P\to B$ be a principal $G$-bundle and $\psi:H\to G$ a homomorphism of topological groups. A $\psi$-structure for $P$ can be defined in two different ways. I am trying to prove their equivalence.
I will give the definitions below. To do so, let us fix models $EH\to BH$ and $EG\to BG$ for the classifying spaces such that $B\psi :BH\to BG$ can be represented by a fibration (where $B\psi$ is the classifying map for the balanced product $EH\times_\psi G$).
First definition: A $\psi$-structure for $P$ is an equivalence class of pairs $(P',\phi')$ consisting of a principal $H$-bundle $P'\to B$ together with a $\psi$-equivariant bundle map $\phi':P'\to P$ where two pairs $(P',\phi')$ and $(P'',\phi'')$ are equivalent if there exists an isomorphism of principal $H$-bundles $f:P'\to P''$ such that $\phi''\circ f=\phi'$
Second definition: A $\psi$-structure for $P$ is for a choice of a classifying map for $P$, $g:B\to BG$, an equivalence class of lifts $h:B\to BH$ of $g$ in terms of $B\psi$, i.e. a map $h:B\to BH$ satisfying $B\psi \circ h=g$. Two lifts being equivalent if they are homotopic by a homotopy $H:B\times I \to BH$ which is a lift of the constant homotopy, i.e. which satisfies $B\psi \circ H=g\circ proj_1$.
It is easy to get for a representative $(P',\phi')$ a lift as in the second definition and vice versa. Namely, pick a classifying map $h'$ for $P'$, homotop $B\psi \circ h'$ into $g$ and lift the homotopy to $BH$. For the other direction, we can take the pullback bundles.
I wasn't able to show that this construction gives a well defined 1-1 correspondence between these definitions, i.e. that it respects the equivalence relations. Can anyone help? Is there a literature reference?