We study categorical properties of the category SWQOS (S$^$U/WQOS, S$^$L/WQOS, resp) of (upper, lower, resp.) semi-continuous quasi-ordered spaces and the subcategory SWPOS (S$^$U/WPOS, S$^$L/WPOS, resp.) of (upper, lower, resp.) semicontinuous partially ordered spaces. We show that the categories S$^$U/WPOS, S$^$L/WPOS and SWPOS are closed under the formation of initial mono-sources in the category TQOS of topological quasi-ordered spaces, and they we mono-topological, complete and cocomplete epireflective subcategories of the category TQOS. We obtain their MacNeille and universal initial completions, as well as those of subcatego.ies S$^$U/WQOS(S$^$L/WQOS) and SWQOS, in which the topologies are T$\_$0/ and T$_1$, respectively.