Ultrafilter functor ($\mathcal{U}$)

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.

The ultrafilter functor is terminal among endofunctors of $\mathbf{Set}$ that preserve finite coproducts. For such an endofunctor $F$ and $x\in F(A)$, the unique natural map sends $x$ to