If I consider the category of finite sets with the monoidal product defined by the cartesian product are there endofunctors which are not monoidal functors? if so I was wondering if there is a classification of endofunctors of this category? i.e. what do all of them look like?
2026-04-24 13:13:17.1777036397
functor between the category of Finite sets which is not monoidal
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There are many examples, for one any constant functor to a set with more than 1 element. Any example you give is inherently not representable because $C(c,-)$ respects products. However, being monoidal here is not enough to guarantee being representable. One such example is given by the constant functor to the empty set.