  If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is
the minimal flag complex $K^f$ on the same vertex set that contains $K$.
Letting $L$ be the set of vertices, there is a sequence of simplicial
inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral
products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline
X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$.
We show that $\Omega f$ and $\Omega f\circ\Omega g$ have right homotopy
inverses and draw consequences. For a flag complex $K$ the polyhedral product
of the form $(\underline{CY},\underline Y)^K$ is a co-$H$-space if and only if
the $1$-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and
$f\circ g$ have right homotopy inverses in this case.
