Keith wrote:

> Think of it this way, the functor \\(\Delta\\) is a category copying machine. It takes a category and copies or *duplicates* it.

> \\[ \Delta: \mathcal{C} \to \mathcal{C} \times \mathcal{C}, \\]

Level slip? I wouldn't say this \\(\Delta\\) is duplicating the category. I'd say it's duplicating each object _in_ the category. Let's call it \\(\Delta_{\mathcal{C}}\\). It has

\[ \Delta_{\mathcal{C}}(c) = (c,c) \]

for each object \\(c\\) in \\(\mathcal{C}\\).

Of course, there is also something that's duplicating the category \\( \mathcal{C}\\). There's a functor

\[ \Delta_{\mathbf{Cat}} : \mathbf{Cat} \to \mathbf{Cat} \times \mathbf{Cat} \]

and we have

\[ \Delta_{\mathbf{Cat}}(\mathcal{C}) = (\mathcal{C}, \mathcal{C}) \]

There's also a functor

\[ \times_{\mathbf{Cat}} : \mathbf{Cat} \times \mathbf{Cat} \to \mathbf{Cat} \]

that takes the product of any two categories:

\[ \times_{\mathbf{Cat}} (\mathcal{C}, \mathcal{D}) = \mathcal{C} \times \mathcal{D} . \]

We can compose these two functors and get a functor

\[ \times_{\mathbf{Cat}} \circ \Delta_{\mathbf{Cat}} : \mathbf{Cat} \to \mathbf{Cat} \]

which does this:

\[ \times_{\mathbf{Cat}} \circ \Delta_{\mathbf{Cat}} (\mathcal{C}) = \mathcal{C} \times \mathcal{C} \]

So, it sends any category \\(\mathcal{C}\\) to the category \\( \mathcal{C} \times \mathcal{C} \\).

Finally, there's a natural transformation from the identity functor

\[ 1_\mathbf{Cat} : \mathbf{Cat} \to \mathbf{Cat} \]

to this composite functor

\[ \times_{\mathbf{Cat}} \circ \Delta_{\mathbf{Cat}} : \mathbf{Cat} \to \mathbf{Cat} \]

To each category \\(\mathcal{C}\\), this natural transformation assigns a functor from \\(\mathcal{C}\\) to \\( \mathcal{C} \times \mathcal{C} \\). And what is this functor? It's

\[ \Delta_{\mathcal{C}}: \mathcal{C} \to \mathcal{C} \times \mathcal{C}. \]

Hey, we've come full circle!