 Pushforward ultrafilter

For an ultrafilter $U$ on $X$ and a function $g:X\to Y$, define $g_{\ast }U=\{B\subseteq Y:g^{-1}[B]\in U\}$. This is an ultrafilter and inherits countable completeness from a countably complete ultrafilter. It is nonprincipal exactly when no single fibre of $g$ belongs to $U$. Restricting $U$ to a member of $U$ before pushing forward preserves these properties.