 Ultrafilter functor

The ultrafilter functor sends a set $A$ to its set $\mathcal{U}(A)$ of ultrafilters. A function $f:A\to B$ acts by pushforward,It preserves finite coproducts: an ultrafilter on $A\bigsqcup B$ contains exactly one summand and is uniquely induced by an ultrafilter on that summand.