I have to admit that this is not really an answer, but rather some sort of meta-answer with some very general remarks which I hope do not bore everyone reading this; it just seems to me that this is necessary to indicate that it is rather misguided, as Yemon already says in the comments and I strongly agree with, to ask such a question if some book introduces elementary number by means of category theory.

Mathematics is all about the nontrivial, unexpected relationships. Category Theory is not really about finding such relationships, but rather about the correct setting, language and *color* some theory is developed. This point of view does not really contradict the hitherto development of category theory into a huge area of mathematics in its own right, full of nontrivial deep theorems; namely because often there is some geometric or whatever background which is our real motiviation. There are ubiquitous examples (model categories, topoi, stacks, $\infty$-categories, ...) which I don't want to elaborate here.

Anyway, as I said, mathematics really starts when something unexpected happens, which does *not* follow from general category theory. For example, the covariant functor $\hom(X,-)$ is always continuous, but when is it also cocontinuous, or respects at least filtered colimits? It turns out that this leads to a natural finiteness condition on $X$, namely we call $X$ then finitely presented. But finally to arrive at the question, $\mathbb{Z}$ is easily seen to be a inital object in the category of rings, but what theorems from category theory are known about initial objects? Well there is nothing to say, expect that every two initial objects are canonical isomorphic, which is just a trivial consequence of the definition. So $\hom(\mathbb{Z},-)$ is easy to describe, but what about the contravariant functor $\hom(-,\mathbb{Z})$? What happens when you plug in $\mathbb{Z}[x,y,z]/(x^n+y^n=z^n)$ for some fixed $n>2$? Does category theory help you to understand this? This example also shows that although the Yoneda-Lemma says that an object $X$ of a category is determined by its functor $\hom(X,-)$, it does not say you anything about the relationship of $X$ with other objects, for example when we just reverse the arrows. Instead, we have to use a specific incarnation of the category and its objects in order derive something which was not there just by abstract nonsense.

Perhaps related questions are more interesting: Which investigations in elementary number theory have led to some category theory (for example, via categorification), which was then applied to other categories as well, thus establishing nontrivial analogies? Or for the other direction, which general concepts become interesting in elementary number theory after some process of decategorification? But in any case, it should be understood that you have to digest elementary number theory before that ...

great. $\endgroup$16more comments