Designing double auctions is a complex problem, especially when there are restrictions on the sets of buyers and sellers that may trade with one another. The goal of this paper is to develop "black-box reductions" from double-auction design to the exhaustively-studied problem of designing single-sided mechanisms. We consider several desirable properties of a double auction: feasibility, dominant-strategy incentive-compability, the still stronger incentive constraints offered by a deferred-acceptance implementation, exact and approximate welfare maximization, and budget-balance. For each of these properties, we identify sufficient conditions on the two one-sided mechanisms - one for the buyers, one for the sellers - and on the method of composition, that guarantee the desired property of the double auction. Our framework also offers new insights into classic double-auction designs, such as the VCG and McAfee auctions with unit-demand buyers and unit-supply sellers.