Representing an isotone map between two bounded ordered sets by principal lattice congruences

For bounded lattices L₁ and L₂, let \({f\colon L_1 \to L_2}\) be a lattice homomorphism. Then the map \({{\rm Princ}(f)\colon \rm {Princ}(\it L_1) \to {\rm Princ}(\it L_2)}\), defined by \({{\rm con}(x,y) \mapsto {\rm con}(f(x),f(y))}\), is a 0-preserving isotone map from the bounded ordered set Princ(L₁) of principal congruences of L₁ to that of L₂. We prove that every 0-preserving isotone map between two bounded ordered sets can be represented in this way. Our result generalizes a 2016 result of G. Grätzer from \({\{0,1}\}\)-preserving isotone maps to 0-preserving isotone maps.