Monad structure on a terminal endofunctor

Let a full subcategory of the endofunctor category contain the identity, be closed under composition, and have terminal object $T$. Then $T$ has a unique monad structure: the unit $1\to T$ and multiplication $T^2\to T$ are the unique maps to the terminal object. The monad laws hold because each compares two maps with codomain $T$ from the same object.

The ultrafilter functor carries the monad structure obtained from its terminality among finite-coproduct-preserving endofunctors of sets. Its unit sends a point to its principal ultrafilter; its multiplication sends an ultrafilter of ultrafilters to the ultrafilter of subsets whose corresponding basic set of ultrafilters is large.