Summary: 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 G. Cherlin et al. [Adv. Appl. Math. 22, No. 4, 454–491 (1999; Zbl 0928.03049)]) and on the other side this leads to the new proof of the existence of dual objects (established by the second author and C. Tardif [J. Comb. Theory, Ser. B 80, No. 1, 80–97 (2000; Zbl 1024.05078)]). Our explicit 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 [Eur. J. Comb. 29, No. 4, 796–820 (2008; Zbl 1160.05024)]).