We study the hyperspace K0(X) of non-empty compact subsets of a Smyth-complete quasi-metric space (X, d). We show that K0(X), equipped with the Hausdorff quasi-pseudometric Hd forms a (sequentially) Yoneda-complete space. Moreover, if d is a T1 quasi-metric, then the hyperspace is algebraic, and the set of all finite subsets forms a base for it. Finally, we prove
that K0(X), Hd) is Smyth-complete if (X, d) is Smyth-complete and all compact subsets of X are d−1-precompact.