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Edit: I have been asked to explain this, so I will. We'll interpret "commutative monoid" in any cocomplete symmetric monoidal category $C$ where $\otimes$ distributes over colimits (each $X \otimes -$ preserves colimits); the simplest way of ensuring that is to assume the category is symmetric monoidal closed.

Then, at the level of formulas, the free commutative monoid is

$$\exp(X) = \sum_{n \geq 0} X^{\otimes n}/\mathbf{n!}$$

where $\mathbf{n!}$ is the categorifier's notation for the symmetric group $S_n$, and we divide out by the canonical action of the $S_n$ on $X^{\otimes n}$.

There is an awful lot more to say about the categorified analogy, but I'll just say one. Using the hypotheses on the symmetric monoidal category $C$, the object $\exp(X)$ carries a commutative monoid structure, and in fact it is the free commutative monoid on the object $X$ (think of the symmetric algebra for the category $C = Vect$, for instance). Like any free functor, the left adjoint $\exp$ preserves colimits, for example coproducts. What is the coproduct of two commutative monoids (in the category of commutative monoid objects)? Their tensor product in $C$! Thus, we arrive at the exponential law

$$\exp(X + Y) \cong \exp(X) \otimes \exp(Y)$$

and this has many applications.