Let us consider a diagram in any suitable( proper) model category as follows:
Where $X' \to X,$ $Y' \to Y$ and $Z' \to Z$ are weak equivalences and P and P' are the pushout of the diagram $Z' \leftarrow X' \to Y'$ and $Z \leftarrow X \to Y$ respectively. Then my questions are the following:
(1) I think there exists a weak equivalence $P' \to P$ such that the diagram commutes. What should be a detailed proof of it?
(2) What is the pushout of the diagram $Z \leftarrow Z' \to P'?$ My guess is $P$ but I don't have any proof for it.
Thank you so much in advance. Any help will be appreciated.
I presume the squares X'Y'XY and X'Z'XZ are commutative in what follows.
(1) is false in general. Additional assumptions are needed, e.g., that both squares XYZP and X'Y'Z'P' are homotopy pushouts. Under such an assumption, (1) becomes a tautology.
(2) is false. Counterexample: all spaces are points, except Y=Δ^1.