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_.

> 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_.