For a broad collection of categories $\cal K$, including all presheaf categories, the following statement is proved to be consistent: every left exact (i.e. finite-limits preserving) functor from $\cal K$ to $\Set$ is small, that is, a small colimit of representables. In contrast, for the (presheaf) category ${\cal K}=\Alg(1,1)$ of unary algebras we construct a functor from $\Alg(1,1)$ to $\Set$ which preserves finite products and is not small. We also describe all left exact set-valued functors as directed unions of ``reduced representables'', generalizing reduced products.
Keywords: left exact functor, small functor, regular ultrafilter.
2000 MSC: 18A35, 18C99, 04A10.
Theory and Applications of Categories, Vol. 8, 2001, No. 13, pp 377-390.
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