I'm reading Homotopy associativiy of H-spaces I by Stasheff and in the proof of Proposition 2 he says that any map is equivalent to a fibring (in modern day language I think this is either a fibration or a fiber bundle). He refers to Lemma 13 of this paper by Nomura, where he states that any map $f:X\to Y$ is equivalent to a fibring. The fibring is of the form $p:Z_f\to Y$ where $Z_f$ is constructed in the proof.
I would like to know in what sense these maps are equivalents. The map $f$ need not be a fibring so this is not an equivalence of fibrings. A homotopy equivalence wouldn't make sense either because the domains are different. I haven't found in any of the linked papers any definition of this equivalence, so does anyone knows what they're talking about?
According to Stashef there should be a map $X\to Z_f$ commuting with $f$ and $p$, so maybe they mean that $f$ is homotopy equivalent to the composition of $p$ with this map.