Eckmann-Hilton duality

Statement
Suppose $$S$$ is a set and $$*$$ and $$\cdot$$ are binary operations on $$S$$ satisfying:

$$\! (a * b) \cdot (c * d) = (a \cdot c) * (b \cdot d) \ \forall a,b,c,d \ \in S$$.

Note that this condition is symmetric in $$*$$ and $$\cdot$$, and can be interpreted as saying that $$\cdot$$ is a homomorphism from the magma $$(S \times S, * \times *)$$ to $$(S,*)$$. In the special case where $$* = \cdot$$, we get what is called a medial magma.

Suppose we have the following additional hypothesis:

Hypothesis: There exists $$e \in S$$ that is a two-sided neutral element for $$*$$ as well as for $$\cdot$$.

Then, we obtain the conclusions:


 * $$* = \cdot$$.
 * The common operation is commutative.