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\$$. 