As John wrote it, I see nothing wrong,

>There's a functor

>$\text{Ob}: \mathbf{Cat} \to \mathbf{Set}$

>that forgets everything about a category except its objects. In other words, it sends any category \$$\mathcal{C}\$$ to its set of objects \$$\mathrm{Ob}(\mathcal{C})\$$, and sends any functor \$$F: \mathcal{C} \to \mathcal{D}\$$ to the function it defines on objects, which I've been calling just \$$F : \mathrm{Ob}(\mathcal{C}) \to \mathrm{Ob}(\mathcal{D})\$$. A more systematic name for it is \$$\mathrm{Ob}: \mathrm{Ob}(\mathcal{C}) \to \mathrm{Ob}(\mathcal{D})\$$.

In fact, John insists very strongly that \$$\mathbf{Cat}\$$ is a category in its own right. We therefore get an even more uncomfortable case,

\$\mathbf{Cat}(\mathrm{Disc}(S),\mathbf{Cat}) \cong \mathbf{Set}(S,\mathrm{Ob}(\mathbf{Cat})). \$