Finite-limit-and-colimit preserving endofunctor of sets
= 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 $A$.