ct.category theory - Why is a monoid with closed symmetric monoidal module category commutative?

(I have rewritten parts to respond to potential objections, since someone disliked this answer. I didn't change the thrust of my discussion, but I did alter some places where I might have been slightly too glib. If I missed something, please feel free to correct me in the comments!)

The situation is even better than that! Suppose we are given an $E_1$-algebra $A$ of a presentable symmetric monoidal $infty$-category $mathcal{C}$.

Call an $E_n$-monoidal structures on the $infty$-category $mathbf{Mod}(A)$ of left $A$-modules allowable if $A$ is the unit and the right action of $mathcal{C}$ on $mathbf{Mod}(A)$ is compatible with the $E_n$ monoidal structure, so that $mathbf{Mod}(A)$ is an $E_n$-$mathcal{C}$-algebra. Then the space of allowable $E_n$-monoidal structures is equivalent to the space of $E_{n+1}$-algebra structures on $A$ itself, compatible with the extant $E_1$ structure on $A$. (This is even true when $n=0$, if one takes an $E_0$-monoidal category to mean a category with a distinguished object.) The object $A$, regarded as the unit $A$-module, admits an $E_n$-algebra structure that is suitably compatible with the $E_1$ structure an $A$. [Reference: Jacob Lurie, DAG VI, Corollary 2.3.15.]

Let's sketch a proof of this claim in the case Peter mentions. Suppose $A$ is a monoid in a presentable symmetric monoidal category $(mathbf{C},otimes)$. Suppose $mathbf{Mod}(A)$ admits a monoidal structure (not even a priori symmetric!) in which $A$, regarded as a left $A$-module, is the unit. I claim that $A$ is a commutative monoid. Consider the monoid object $mathrm{End}(A)$ of endomorphisms of $A$ as a left $A$-module; the Eckmann-Hilton argument described below applies to the operations of tensoring and composing to give $mathrm{End}(A)$ the structure of a commutative monoid object. The multiplication on $A$ yields an isomorphism of monoids $Asimeqmathrm{End}(A)$.

In the case you mention, the result amounts to the original Eckmann-Hilton, as follows. If $X$ admits magma structures $circ$ and $star$ with the same unit (Below, Tom Leinster points out that I only have to assume that each has a unit, and it will follow that the units are the same. He's right, of course.) with the property that

$$(acirc b)star(ccirc d)=(astar c)circ(bstar d)$$

for any $a,b,c,din X$, then (1) the magma structures $circ$ and $star$ coincide; (2) the product $circ$ is associative; and (3) the product $circ$ is commutative. That is, a unital magma in unital magmas is a commutative monoid.

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