  This paper unifies problems and results related to (embedding) universal and
homomorphism universal structures. On the one side we give a new combinatorial
proof of the existence of universal objects for homomorphism defined classes of
structures (thus reproving a result of Cherlin, Shelah and Shi) and on the
other side this leads to the new proof of the existence of dual objects
(established by Ne\v{s}et\v{r}il and Tardif). Our explicite approach has
further applications to special structures such as variants of the rational
Urysohn space. We also solve a related extremal problem which shows the
optimality (of the used lifted arities) of our construction (and a related
problem of A. Atserias).
