Ordering the order of a distributive lattice by itself

We order the ordering relation of an arbitrary poset P component-wise by itself, obtaining a poset Φ(P) extending P. In particular, the effects of Φ on L ∈ DLAT01, the category of all bounded distributive lattices, are studied, mainly with the aid of Priestley duality. We characterize those L ∈ DLAT01 which occur as Φ(K) for some K ∈ DLAT01, decide this situation in polynomial time for finite L, characterize fixpoints of Φ within DLAT01 and relate them to free objects in DLAT01.