Finite-limit-and-colimit preserving endofunctor of sets

An endofunctor $F:\mathbf{Set}\to \mathbf{Set}$ preserving finite limits and finite colimits has a unique natural monomorphism $\alpha :1_{\mathbf{Set}}\to F$. It also preserves countable coproducts. If some ${\alpha }_A$ is not surjective, the extra element determines a countably complete nonprincipal ultrafilter on $A$.

For an ultrafilter $U$ on $A$, the assignmentis an endofunctor of sets. It preserves finite limits. If $U$ is countably complete, every map $A\to {\coprod }_n{B}_n$ lands in one summand on a $U$-large set, so $F$ also preserves countable coproducts.