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?