Glueing lemma in model category

Question

Everything below takes places in some model category and all objects are assumed to be cofibrant.

$\require{AMScd}$

Fact 1 Given a pushout $$\small{\begin{CD} A @>i>> B\\ @VfV\simeq V @VVgV\\ C @>>>D \end{CD}}$$ with $i$ a cofibration and $f$ a weak equivalence, then $g$ is also a weak equivalence.

Fact 2 Given a map of diagrams $$\small{\begin{CD} B @<<< A @>>> C\\ @V\simeq VV @V\simeq VV @V\simeq VV\\ B' @<<< A' @>>>C' \end{CD}}$$ with all horizontal maps cofibrations, then the induced maps of pushouts is a weak equivalence.

How do I use these facts to deduce the analogue of fact 2 where we only require the left horizontal maps to be cofibrations?

Answer 1

Factor the right hand maps into cofibrations with codomains $D,D'$ followed by trivial fibrations, so that the pushout cube you had is split into two cubes, with back-right arrows $E\to E'$ and $P\to P'$. The former is a weak equivalence by Fact 1. $D\to E$ and $D'\to E'$ are pushouts of cofibrations, thus cofibrations. $E\to P$ and $E'\to P'$ are weak equivalences because they're pushouts of weak equivalences (in particular, trivial fibrations) along cofibrations. Thus $P\to P'$ is a weak equivalence by 2-of-3.

For what it's worth, the most appropriate setting for this result is in a left proper model category. We're just using the fact that the subcategory of cofibrants is always left proper.