Owen wrote:

> This reminds me of the fact that if \\(M\\) is an ordinary monoid, then the multiplication function \\(M\times M\to M\\) isn't a monoid homomorphism unless \\(M\\) is commutative. Is that just a coincidence, or is there something deeper connecting both of these ideas?

Category theorists have a slick way to express the fact you're mentioning, and a bit more: _a monoid in the category of monoids is exactly the same as a commutative monoid._

The proof is called the ['Eckmann-Hilton argument'](https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument#Remarks_2), and it was first used to note that _a group in the category of groups is exactly the same as an abelian group_.

It admits lots of generalizations. It's _related_ to the fact that Anindya mentioned: _the category of categories enriched over a symmetric monoidal preorder is itself monoidal_. But the relation is a bit elusive.

It becomes slightly clearer (?) if we push the Eckmann-Hilton argument up one level, and also replace preorders by categories. Then we have these nice facts:

1. A monoid in the category of monoidal categories is a braided monoidal category.

2. The category of \\(\mathcal{V}\\)-enriched categories is monoidal if \\(\mathcal{V}\\) is a braided monoidal category.

These statements are closely connected. But I can't quite put my finger on the connection.

I could go on, listing more facts that illustrate this connection, but this is probably enough for now!