Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ L denote the ordered set of principal congruences of L. For {0,1}-sublattices A subseteq B of L, congruence generation defines a natural map from Princ A to Princ B. In this way, we obtain a small category of bounded ordered sets as objects and some 0-separating {0,1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: each small category of bounded ordered set with these properties is representable by principal lattice congruences in the above sense.