Jesus wrote:

> What is exactly "\\(\mathbf{2}\\)"? In puzzle 151 I think it is just two isolated identities. But here it is as in [this comment](https://forum.azimuthproject.org/discussion/comment/19672/#Comment_19672). So the category of \\(\mathcal{C}\\)-instances is more exactly \\(\mathbf{Set}^{\rightarrow}\\), the [arrow category](https://ncatlab.org/nlab/show/arrow+category) of **Set**, that is, the category of functions and commutative squares.

I've been using \\(\mathbf{2}\\) with a boldface to mean the category you're calling \\(\rightarrow\\), the category with two objects and one non-identity morphism. So \\(\mathbf{Set}^{\mathbf{2}}\\) is just what you say: the arrow category of \\(\mathbf{Set}\\), whose objects are functions and whose morphisms are commutative squares.

On the other hand, in Puzzle 151 I wrote \\(\mathbf{Set}^2\\) - no boldface on the \\(2\\) - to stand for the category \\(\mathbf{Set} \times \mathbf{Set}\\), whose objects are pairs of sets and whose morphisms are pairs of functions. Sorry! It's hard to read the difference in font. But yes, if we wanted, we could use \\(2\\) without a boldface to mean the category with two objects and only identity morphisms. Then \\(\mathbf{Set}^2\\) would be another example of a functor category.

But yeah, it's bad use of notation to have \\(\mathbf{Set}^{\mathbf{2}} \ne \mathbf{Set}^2\\).