On characteristics of homomorphism and embedding universal graphs

We relate the existence problem of universal objects to the properties of corresponding enriched categories (lifts and expansions). Particularly, extending earlier results, we prove that for every countable set F of finite structures there exists a (countably) universal structure U for the class Forbh(F) (of all countable structures omitting a homomorphism from all members of F). In fact U is the shadow (reduct) of an ultrahomogeneous structure U (which however, as we will show, cannot be expressed as Forbh(F ) for a countable set F ; this is in a sharp contrast to the case when F is finite). We also put the results of this paper, perhaps for the first time, in the context of homomorphism dualities and Constraint Satisfaction Problems.