  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 $K_2.$
Applications are given to a family universal $H$-colourable graphs, for $H$ a
finite core.
