Realizing the monoid $\mathbb{N}/(3=1)$ from a category with finite coproducts If $\mathcal{C}$ is a category with finite coproducts (including an initial object $0$), then the set of isomorphism-classes of $\mathcal{C}$ becomes a commutative monoid with $0 := [0]$ and $[x] + [y] := [x \oplus y]$. Is there a category $\mathcal{C}$ with finite coproducts such that the associated commutative monoid of isomorphism-classes is isomorphic to $\mathbb{N}/(3=1)$? Notice that the commutative monoid $\mathbb{N}/(3=1)$ has three elements $0,1,2$ and that we compute with $2n+1=1$ and $2n+2=2$ for $n \geq 1$.
In other words, there are exactly three non-isomorphic objects $0,a,b$, such that $b$ is a coproduct of $a$ with itsself and $a$ is a coproduct of $b$ with $a$. This requires the existence many morphisms, for example four coproduct inclusions, but also an involution $b \to b$ which permutes the two copies of $a$. In order to get the universal properties, we probably need even more morphisms, and I don't know where to stop. My idea is to construct a universal example, maybe it is more easy to construct a more specific example where some of the mentioned morphisms agree.
A more general question would be which commutative monoids arise as monoids of isomorphism-classes of categories with finite coproducts. But in order to get started, let us look at some examples first.
 A: The following construction is due to P.M. Cohn ("Some remarks on the invariant basis property", Topology 5, 215-228 (1966)) who developed the idea into the ring construction called "universal localization". There are previous, more complicated examples, but this one is probably more easily generalizable.
Let $R$ be the ring generated by elements $x_1,x_2,x_3,y_1,y_2,y_3$ subject to the relations $x_1y_1+x_2y_2+x_3y_3=1$ and $y_ix_j=\delta_{ij}$ (i.e., exactly the relations that are needed to make the matrices $\left(\begin{array}{ccc}x_1&x_2&x_3\end{array}\right)$ and $\left(\begin{array}{c}y_1\\y_2\\y_3\end{array}\right)$ inverse to one another).
Then these matrices give $R$-module isomorphisms between $R$ and $R^3$.
It's maybe not entirely obvious that $R\not\cong0$ or that $R\not\cong R^2$, but these facts are proved in Section 5 of Cohn's paper.
This is the "universal" example of an additive category with the monoid of isomorphism classes you want. If a given monoid is realizable by an additive category, then I think the additivity makes it easier to construct examples, since you can express the fact that one object is the coproduct of two others in terms of simple equations involving maps. I don't know if there could be monoids realizable by a non-additive category but not by an additive one?
