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If $\mathcal{A}$ and $\mathcal{B}$ are small categories (i.e. objects are sets, not proper classes) and $F,G:\mathcal{A} \to \mathcal{B}$ are both (contra/co)variant functors, then the the 'collection' of natural transformations $Nat(F,G)$ is also a set.

Say we now have arbitary categories $\mathcal{C}$ and $\mathcal{D}$, with functors $S,T$ such that $S,T:\mathcal{C} \to \mathcal{D}$. Then it seems logical that $Nat(S,T)$ may be a proper class, and not a set.

Can anyone provide any examples?

Moreover (this is a bit more philosophical) - are such examples pathological, or do non-small categories arise naturally, outside of say, just studying higher category theory?

I am guessing in something like higher homotopy theory, one could possibly get in trouble?

This thread seems somewhat related

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    "objects are sets, not proper classes" is not what "small category" means. It means the class of all objects is a set, not that each object is one.2011-06-03
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    @Alon - thank you for that. Is that why the category of sets is not a small category - there is no collection of all sets (in ZF set-theory)?2011-06-03
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    Yes. The same goes for the other common categories Mariano mentions - there's no set of all groups, no set of all topological spaces, etc.2011-06-03

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