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For a lattice L with 0 and 1, let Princ L denote the ordered set of principalcongruences of L. For {0,1}-sublattices A subseteq B of L, congruencegeneration defines a natural map from Princ A to Princ B. In this way, weobtain a small category of bounded ordered sets as objects and some0-separating {0,1}-preserving monotone maps as morphisms such that everyhom-set consists of at most one morphism. We prove the converse: each smallcategory of bounded ordered set with these properties is representable byprincipal lattice congruences in the above sense.

Representing some families of monotone maps by principal lattice congruences