A naturally occurring non-locally small category

Let $\mathcal{C}$ be a category. We say that $\mathcal{C}$ is locally small if $\mathrm{Hom}_{\mathcal{C}}(A,B)$ is a set for all $A$, $B$ in $\mathcal{C}$.

I can't think of any natural examples of non-locally small categories which are 'obviously' not locally small. We can take $\mathcal{C}$ to have one object, and a morphism for every $x \in V$ (say in ZFC), with composition of morphisms given by the union of two sets, but I can't think when this would ever come up 'naturally'

Are there any natural examples of non-locally small categories which are obviously not locally small?

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–  Julian Kuelshammer Oct 23 '12 at 18:04
In general, the functor category $\mathbf C^{\mathbf D}$ is not locally small if $\mathbf C$ and $\mathbf D$ are large. –  Lord_Farin Oct 23 '12 at 18:07
You done stole my idea :( I suppose the example comes up naturally by turning the (set theoretic) universe into a monoid under $\cup$. –  Clive Newstead Oct 23 '12 at 18:25
There are categories that are locally small but not obviously so – the category obtained by formally inverting all weak homotopy equivalences between topological spaces, for example. –  Zhen Lin Oct 23 '12 at 18:39