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$L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$. This induces a sequence of maps of polyhedral products$(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$. We show that$\unicode[STIX]{x1D6FA}f$and$\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$have right homotopy inverses and draw consequences. For a flag complex$K$the polyhedral product of the form$(\text{}\underline{CY},\text{}\underline{Y})^{K}$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.