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})).

\\]

>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})).

\\]