Igor wrote:

> Btw, \$$fg = gf\$$ implies that composition of morphisms in \$$\mathcal{C}(I, I)\$$ is commutative, it is interesting whether there are other implications for \$$I\$$.

Not much:

**Theorem.** For any commutative monoid \$$A\$$, there is a monoidal category \$$\mathcal{C}\$$ with just one object \$$I \$$ (the unit for the tensor product) such that \$$\mathcal{C}(I,I) = A\$$, with both tensor product and composition equal to the multiplication in \$$A\$$.

In simpler, rougher terms: _a one-object monoidal category is the same as a commutative monoid_.