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Suppose $\mathcal{F},\mathcal{G}:\mathcal{C}\rightarrow\mathcal{D}$ are functors and $\mathcal{C},\mathcal{D}$ are locally small categories. Is it true that the collection of natural transformations from $\mathcal{F}$ to $\mathcal{G}$, $\text{Nat}(\mathcal{F},\mathcal{G})$, is a set?

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  • $\begingroup$ $\mathcal{C}$ needs to be legitimately small: a set of objects, not a proper class. $\endgroup$ – Randall Nov 3 '17 at 19:34
  • $\begingroup$ Yes. And $\mathcal{D}$ should be at least locally small I think $\endgroup$ – piotrmizerka Nov 3 '17 at 19:38
  • $\begingroup$ ^Also true, yes. $\endgroup$ – Randall Nov 3 '17 at 19:38
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No; in fact, $\text{Nat}(\mathcal{F},\mathcal{G})$ may even be "larger than a proper class" (though what this means exactly depends on what set theoretic framework you are using). For instance, let $\mathcal{C}=\mathcal{D}$ have a proper class of objects, no morphisms between distinct objects, and a cyclic group of order $2$ of endomorphisms of each object. Taking $\mathcal{F}=\mathcal{G}$ to be the identity, a natural transformation $\mathcal{F}\to\mathcal{G}$ consists a choice of one of the two endomorphisms on each object. There is certainly not a set of such choices. If you work with universes and say $\mathcal{C}$ has $\kappa$ objects for some inaccessible cardinal $\kappa$, then there will be $2^{\kappa}$ ways to choose a natural transformation $\mathcal{F}\to\mathcal{G}$.

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