Definition of natural numbers in category theory

Question

It is said that category theory serves as an alternative foundation of mathematics, as such it must define natural numbers in terms of categories as it is done in set theory, when we consider set theory as a foundation for mathematics and where we define recursively natural numbers as $0:=\emptyset$ and $n+1:=n\cup \{n\}$. So, what is the definition of natural numbers in category theory?

Answer 1

I'm sure there are plenty, but this is the one that I have encountered, I believe it is rather standard.

In the category of small categories, consider the category C with one object and only the identity morphism, and the category D with two objects, a single morphism between the two objects, the two identity morphisms, and no other morphisms. Then there are two functors $C\to D$. The natural numbers $\Bbb N$ is the coequalizer of these two functors.

This coequalizer $\Bbb N$ has a single object, and there is a coequalizing functor $D\to \Bbb N$. The morphism between the two objects in $D$ is mapped to the morphism $1$ in $\Bbb N$, and apart from the identity morphism in $\Bbb N$, all other morphisms can be obtained in a unique way by composing $1$ with itself. Composition is the addition operation. (Note that this construction is mentioned in the "Examples" section of the coequalizer Wikipedia article.)