Homomorphisms and principal congruences of bounded lattices I. Isotone maps of principal congruences

Two years ago, I characterized the order Princ L of principal congruences of a bounded lattice L as a bounded order. If K and L are bounded lattices and phi is a {0, 1}- homomorphism of K into L, then there is a natural isotone {0, 1}- map Princ phi from Princ K into Princ L. We prove the converse: For bounded orders P and Q and an isotone {0, 1}- map of psi into Q, we represent P and Q as Princ K and Princ L for bounded lattices K and L with a {0, 1}- homomorphism psi of K into L, so that is represented as Princ phi.