I remember the following definition of a homotopy fibration. If we have a continuous map $f:X\to Y$, then there exists a homotopy equivalence $h:X\to P_f$ and a (Hurewicz) fibration $p:P_f\to Y$ such that $f=p\circ h$. Let $F$ be a fiber of $p$. In this case, we call the space $F$ a homotopy fiber of the map $f$ (and denote it by $\operatorname{hofib}(f)$) and we call $$F\xrightarrow{g} X\xrightarrow{f} Y$$ a homotopy fibration. Furthermore, the space $P_f$ (the mapping path space) is constructed explicitly as a subspace of $X\times PY$, which allows us to write explicit formulae for $h$ and $p$.
Also, as far as I understand, if we have any other factorization $f=p'\circ h'$, the fiber of $p'$ is homotopy equivalent to $\operatorname{hofib}(f)$, i.e., the homotopy fiber is defined correctly. Thus, if we have a homotopy equivalence $h':X\to \widetilde{X}$ and a fibration $p':\widetilde{X}\to Y$ with fiber $F'$, then we call $$F'\xrightarrow{g'} X\xrightarrow{f} Y$$ a homotopy fibration too, despite the space $\widetilde{X}$ not being the mapping path space.
My question is, what are the maps $g$ and $g'$ in these cases explicitly? The definition itself says nothing about them, so I guess the answer cannot be deduced, and my definition is just not complete.
My guess.
In the case of $$F\xrightarrow{g} X\xrightarrow{f} Y$$ I have been told that $g$ it the projection $(x,\gamma)\mapsto x$.
In the case of $$F'\xrightarrow{g'} X\xrightarrow{f} Y$$ I assume it would be logical if the map $g'$ were the composition $F'\xrightarrow{i} \widetilde{X} \xrightarrow{h''} X,$ where the left arrow $i$ the inclusion of fiber of fibration $p':\widetilde{X}\to Y$ into its total space $\widetilde{X}$, and $h''$ is the homotopy inverse of $h'$.
However, if I am right in the case of $$F'\xrightarrow{g'} X\xrightarrow{f} Y,$$ is seems to me that it would be more natural if in the case of $$F\xrightarrow{g} X\xrightarrow{f} Y$$ we had $g: F\xrightarrow{i} P_f \xrightarrow{H} X$, where $H$ is the homotopy inverse of the (explicitly constructed) homotopy equivalence $h$ between the mapping path space and $X$ from the definition of a homotopy fibration, and $i$ is inclusion of fiber of fibration $P_f \to Y$ into its total space. Hence, I am a bit lost.
Thank you.