Does the category with two objects, one arrow exist?

Question

According to pg. 10 of Categories for the Working Mathematician, there exists a category called $\mathbf{2}$ s.t.

$\mathbf{2}$ is the category with two objects $a$, $b$ and just one arrow $a \rightarrow b$ not the identity

But doesn't this violate the axiom that all categories must have, for each object, an identity arrow? Here there is no identity arrow $id_a$ and $id_b$, which seems to suggest that $\mathbf{2}$ is in fact not a category.

Answer 1

This is a wording issue. The phrase "just one arrow $a\rightarrow b$ not the identity" means that the category has only one non-identity arrow; so this is a category with three arrows total, the identities on $a$ and $b$ and the unique $a\rightarrow b$. (Note that the non-identity arrows determine the entire category!)

It would probably be clearer to write it as "just one arrow which is not an identity", but oh well.