Summary: Burke proved that the generalized nonstationary ideal, denoted by NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of NS to some stationary set. We investigate how far Burke’s theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \(\mathcal{C}\)-systems of filters introduced by Audrito and Steila. First we answer a question of G. Audrito and S. Steila [J. Symb. Log. 82, No. 3, 860–892 (2017; Zbl 1422.03111)], by proving that \(\mathcal{C}\)-systems of filters do not capture all kinds of set-generic embeddings. We provide a characterization of supercompactness in terms of short extenders and canonical projections of NS, without any reference to the strength of the extenders; as a corollary, NS can consistently fail to canonically project to arbitrarily strong short extenders. We prove that \(\omega \)-cofinal towers of normal ultrafilters, e.g., the kind used to characterize \(\mathrm{I2}\) and \(\mathrm{I3}\) embeddings, are well-founded if and only if they are canonical projections of NS. Finally, we provide a characterization of “\(\aleph_{\omega}\) is Jónsson” in terms of canonical projections of NS.