The $n$Lab writes (prop. 2.2 in <https://ncatlab.org/nlab/show/category+of+monoids>) :


Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the forgetful functor $U\_C$ has a left adjoint $F\_C:C\rightarrow Mon(C)$. 


Suppose now $C$ is a symmetric monoidal category. Under what conditions does the canonical forgetful functor $U\_C:CMon(C)\rightarrow C$ from the category of commutative monoid objects in $C$ have a left adjoint? How is this left adjoint constructed? 



Left adjoint for categories of commutative monoids?
The left adjoint is the functor Sym, defined as $Sym(X) = I \coprod X \coprod (X \otimes X)/\Sigma\_2 \coprod (X\otimes X\otimes X)/\Sigma\_3 \coprod \dots$, where $I$ is the unit. I only ever work in closed, cocomplete settings, where this always exists, but in general I think it should exist under the same conditions as the free monoid functor, as long as the colimits $X^{\otimes n}/\Sigma\_n$ exist.


