For a non-empty space \(X\) and a non-trivial lattice \(Y\), some connections between properties of the poset \([X,Y]\) of all continuous functions \(X\to Y\) and properties of \(X\) and \(Y\) are well known. For instance, \([X,Y]\) is a continuous lattice if, and only if, both \(Y\) and \(\mathcal {O}X\) are continuous lattices. In the article this result is extended to certain classes of \(\mathcal {Z}\)-distributive lattices, where \(\mathcal {Z} \) is the Wright-Wagner-Thatcher subset system. While this is likely to be the main aim of the article, other results of a similar nature are obtained as well.