 Eckmann–Hilton argument

If two unital operations on one set share their identity and satisfy $(a\ast b)\cdot (c\ast d)=(a\cdot c)\ast (b\cdot d)$, then the operations coincide and are commutative.

The Eckmann–Hilton argument is a concept in category theory and homotopy theory that plays a role in the context of algebraic structures such as monoids and operads. It particularly addresses the interactions between two operations defined on a space or an algebraic structure when these operations are defined in a certain way, especially in relation to commutativity and associativity.