Distinguishing homomorphisms of infinite graphs

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper n-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph G satisfies the connected existentially closed property and admits a homomorphism to H, then it admits continuum-many distinguishing homomorphisms from G to H join K2. Applications are given to a family universal H-colourable graphs, for H a finite core.