How large are left exact functors? (Q2724147)

\textit{A. Blass} [``Exact functors and measurable cardinals'', Pac. J. Math. 63, 335-346 (1976; Zbl 0348.18007)] proved that every left exact functor \({\mathcal S}et\to {\mathcal S}et\) is small (i.e., a small colimit of hom-functors) provided that every uniform ultrafilter on an infinite set is regular. \textit{H.-D. Donder} [``Regularity of ultrafilters and the core model'', Isr. J. Math. 63, No. 3, 289-322 (1988; Zbl 0663.03037)] showed that this assumption is consistent with ZFC. The authors extend it to all left exact functors \({\mathcal S}et^{\mathcal A}\to {\mathcal S}et\) for any small category \(\mathcal A\). This solves the problem recently put by the first author, \textit{F. W. Lawvere} and the reviewer asking to the legitimacy of small-ary operations on locally finitely presentable categories. One should add that \textit{J. Reiterman} found a non-small left exact functor \({\mathcal S}et\to {\mathcal S}et\) assuming the existence of a proper class of measurable cardinals. NEWLINENEWLINENEWLINEThe situation changes if one considers functors \({\mathcal S}et^{\mathcal A}\to {\mathcal S}et\) preserving finite products. The authors construct a large functor of this kind for the monoid \(\mathcal A\) on two generators.