TITLE: Fibre preserving maps of Borel constructions
QUESTION [6 upvotes]: Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times_G X\to EG\times_G Y$ between Borel constructions, as in the following diagram.
$\require{AMScd}$
\begin{CD}
    X @>\overline{f}>> Y\\
    @V  V V @VV V\\
    EG\times_G X  @>>f> EG\times_G Y \\
     @V  V V @VV V\\
    BG @>=>> BG
\end{CD}
I believe some sort of converse to be true. That is, if $f:EG\times_G X\to EG\times_G Y$ is a fibre-preserving map between Borel constructions, it induces a $G$-equivariant map $\overline{f}:X\to Y$. Here I am identifying $G=\pi_1(BG,\ast)$, which acts (up to homotopy) on the fibres.

Does anyone know of a reference to a precise statement along these lines?

Edit (in response to user51223's comment): I'm mainly interested in the existence question: Is it true that there exists an equivariant map $\overline{f}:X\to Y$ if and only if there exists a map $f:EG\times_G X\to EG\times_G Y$ over $BG$? (So $\overline{f}$ doesn't have to be the induced map on fibres.)

REPLY [2 votes]: Similar to Granja's answer: let $X$ be a point, and $Y = EG$, and let $f$ be induced by the diagonal on $EG$.