Ultrafilter limit map for a dense continuous image ($\Lambda (\mathcal{H})=\lim f_{\ast }\mathcal{H}$)

A continuous dense map into a compact Hausdorff space assigns to each ultrafilter on its domain the unique limit of its pushforward. This limit map is surjective by the ultrafilter characterization of image closure. It is injective exactly when disjoint domain subsets have disjoint image closures.