As discussed in the comments under this answer:
What choice of morphisms on the category of ordinals yields Hessenberg arithmetic, i.e. the Hessenberg sum as the categorical coproduct and the Hessenberg product as the categorical product?
Furthermore, if the morphisms are order-preserving maps, do coproducts or products exist?
The ordinals under Hessenberg arithmetic are the free commutative semiring on ordinals of the form $\omega^{\omega^\alpha}$ - any ordinal can be written in Cantor normal form as a sum of powers of $\omega$, then write those powers in Cantor normal form, and rewrite $\omega^{\omega^\alpha+\omega^\beta} \to \omega^{\omega^\alpha}\omega^{\omega^\beta}$ to get a sum of products of powers of powers of $\omega$.
So, take the free distributive category with objects $\omega^{\omega^\alpha}$. The commutative semiring of (isomorphism classes of) objects of this category, with coproduct as addition and product as multiplication (this is a commutative semiring because the category is distributive), is isomorphic to the free commutative semiring on $\omega^{\omega^\alpha}$, which (by the first paragraph) is isomorphic to the commutative semiring of ordinals with Hessenberg arithmetic, so this is the category of ordinals we're looking for.
