Let D have small hom sets. Does this functor category Set^D have small hom-sets? I think this is not true, I mean each functor can't possibly be coded as a small set if D itself is large. I'm working within a universe.
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This is indeed not true. Peter Freyd and Ross Street proved in the paper on the size of categories that both $\mathcal{D}$ and $\mathbf{Set}^{\mathcal{D}}$ have small hom-sets if and only if the category $\mathcal{D}$ is essentially small, so any category which has a large set of objects will give a counterexample.
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$\begingroup$ To be nitpicky, any category with a large set of objects which is ALSO not equivalent to one with a small set of objects would provide a counterexample, e.g. the category of Sets (or more generally, any Grothendieck topos). $\endgroup$ May 24, 2010 at 19:17
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$\begingroup$ or, a discrete category with a large set of objects. $\endgroup$ May 24, 2010 at 19:32
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$\begingroup$ Not the first time the question has been asked. See "Is the presheaf category of a locally small category locally small?" $\endgroup$– Todd Trimble ♦May 24, 2010 at 20:40