KöMaL 2008 A464

Let $H$ a set and denote by $P$($H$) the family of all subsets of $H$. Suppose that $f$ and $g$are $P$($H$)$\to P$($H$) maps such that $f$($X$)$\subset f$($Y$)$\subset H$ and $g$($X$)$\subset g$($Y$)$\subset H$whenever $X \subset Y \subset H$. Prove that there exist sets $A$,$B \subset H$ for which $f$($A$)=$H$$B$and$g$($B$)=$H$\$A$.