If K is a simplicial complex on m vertices the flagification of K is the minimal flag complex K' 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'.

This induces a sequence of maps of polyhedral products (X,A)^L --g--> (X,A)^K --f--> (X,A)^K'.

We show that the loops on f and the loops on f composed with g have right homotopyinverses and draw consequences. This is joint work with Taras Panov. 

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