Typically, when defining the functor category $\mathcal{D}^\mathcal{C}$, where objects are functors $\mathcal{C}\rightarrow\mathcal{D}$ and the morphisms between such objects $F,G$ are the natural transformations $F\rightarrow G$ with the obvious composition and identities, one requires the category $\mathcal{C}$ to be *small*, i.e. that its objects form a set. $\mathcal{D}$ can be any category.

I've always thought that the reason for the smallness requirement on $\mathcal{C}$ is that "surely you'll get into trouble with the morphisms $F\rightarrow G$ in $\mathcal{D}^\mathcal{C}$ potentially not forming a set otherwise", and I've never given it much thought.

Now, does anyone know of an *example* which fixes a large category $\mathcal{C}$, a category $\mathcal{D}$ and two functors $F,G:\mathcal{C}\rightarrow\mathcal{D}$ such that that the collection of natural transformations $F\rightarrow G$ does not form a set? I've always thought in the back of my head that this should be easily doable with $\mathcal{C}=\mathcal{D}=\mathrm{Set}$ and some simple functors $F,G:\mathcal{C}\rightarrow\mathcal{D}$ by somehow cooking up at least one natural transformation $F\rightarrow G$ for each object of $\mathcal{C}$, i.e. for each set. Maybe my set theory fu is just weak.

I could of course just be missing something basic here. I'm asking because the books I've checked tend to just state $\mathcal{C}$ must be small. Apologies if this is all trivial!