I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $\geq \omega$.
Through the basic notions of category theory, what is an ordinal? And what does ordinal arithmetic look like categorically?