  We establish a general method to produce cofibrant approximations in the
model category $U_S(C,D)$ of $S$-valued $C$-indexed diagrams with $D$-weak
equivalences and $D$-fibrations. We also present explicit examples of such
approximations. Here, $S$ is an arbitrary cofibrantly generated simplicial
model category and $D\subset C$ are small categories. An application to the
notion of homotopy colimit is presented.
