Duoidal Category

A duoidal category is a Monoidal Category $(\mathcal{V},\diamond,I)$ equipped with a further monoidal structure $(\star,J)$ such that $\star:\mathcal{V}\times\mathcal{V}\to\mathcal{V}$ and $J:1\to\mathcal{V}$ are both Lax Monoidal Functors with respect to $(\diamond,I)$ and the coherence axioms of $(\star,J)$ are Monoidal Natural Transformations with respect to $(\diamond,I)$.

If we unfold this, we see that:

• Note taken on [2025-04-05 Sat 16:16]
  This is a bit sketchy; check the directions!

• Laxity of $\star$ yields a natural transformation $(A\star B)\diamond(C\star D)\to(A\diamond C)\star(B\diamond D)$ and $I\to I\star I$

• Laxity of $J$ yields maps $J\diamond J\to J$ and $I\to J$

Along with a bunch of axioms that say that $J$ is a $\diamond$-monoid and $I$ is a $\star$-comonoid.

Intuition

Note the remarkable similarity to the setup found in the Eckmann-Hilton Argument; the only difference is that everything is lax.