Let \begin{array}{ccc} X & \xrightarrow{} &Y \\ \downarrow & & \downarrow \\ Z & \xrightarrow{} & W\end{array} be a commutative diagram in a proper model catgeory and $P$ be the homotopy pushout of $Z \leftarrow X \rightarrow Y$. Consider a commutative diagram \begin{array}{ccc} P' & \xrightarrow{\sim} &P \\ \downarrow & & \downarrow \\ W' & \xrightarrow{\sim} & W\end{array} Then is there any way to construct objects $X', Y',$ and $Z'$ in the model category such that we have a following commutative diagram
where $X' \to X, Y' \to Y , Z' \to Z$ and $W' \to W$ are weak equivalences.
Thank you in advance. Any help will be appreciated.
There's no reason you should be able to find even $Y'$ providing the front-right face of your prism. In simplicial sets, Let $Y$ be nonempty and suppose $Y\times_W W'=\emptyset$. Then there exists no map whatsoever $Y'\to W'$ with nonempty domain such that the composite $Y'\to Y$ factors through $Y$, whether by a weak equivalence or not, since any such $Y'$ admits a map to the pullback $Y\times_W W'$.