Eckmann-Hilton intuition for duoidal categories

Intuition

By the Eckmann-Hilton argument, each time we have two monoids $(\ast ,\circ )$ such that one is a monoid homomorphism over the other, $(a\circ b)\ast (c\circ d)=(a\ast c)\circ (b\ast d)$, we know that both monoids coincide into a single commutative monoid.

However, an extra dimension helps us side-step the Eckmann-Hilton argument. If, instead of equalities or isomorphisms, we use directed morphisms, both monoids (which now may become 2-monoids) do not necessarily coincide, and the resulting structure is that of a duoidal category.