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This looks pretty obvious, but an endofunctor $F : Set \to Set$ seems to be a presheaf over $Set^{op}$.

Is there any useful fact that can be learnt from this view of endofunctors?

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We usually only consider presheaves over small categories. $\mathbf{Set}$ is not small. –  Zhen Lin Nov 22 '13 at 16:09
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Perhaps one should restrict to $F$ preserving directed colimits, or equivalently to functors $F : \mathsf{FinSet} \to \mathsf{Set}$. This is related to combinatorial species. –  Martin Brandenburg Nov 23 '13 at 0:22
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