We show, for any subset system Z (as defined in Wright, Wagner, and Thatcher, T.C.S. 7 (1978), pp. 57–77) and any order preserving map $f:Q \to P$ of posets, the existence of a universal map $u_f :P \to P_f $ where $P_f $ is Z-complete and $u_f f$ is Z-continuous. This generalizes to arbitrary subset systems the result of Markowsky (T.C.S. 4 (1977), pp. 125–135) for chains, and the completions of Wright, Wagner, and Thatcher for union complete Z; our method, different from theirs, uses the time-honored direct construction of universal maps. Further, we obtain some results on the internal structure of $P_f $ with regard to Z-joins. Finally, we show that each element of the Z-completion of P is a Z-join of elements of P iff Z is union complete.