Let $(\mathcal C, \otimes, e, \alpha:\_\otimes (\_\otimes \_)\to (\_\otimes\_)\otimes \_, \lambda :e\otimes \_ \to 1_\mathcal C, \rho : \_\otimes e\to 1_\mathcal C, \gamma: \_\otimes = \to =\otimes \_)$ be a symmetric monoidal category.
Suppose we are given a strong monad $\mathbb T \equiv (T, \eta:1_\mathcal C\to T, \mu : T^2\to T)$ with left-strength $\sigma:\_\otimes T\_\to T(\_\otimes \_)$ (which induces a right-strength $\tau\equiv T\gamma\circ \sigma \circ \gamma:T\_\otimes \_\to T(\_\otimes \_)$).
Recall that $\mathbb T$ is a commutative monad if for any objects $X,Y$, we have that $\mu\circ T\sigma_{X,Y}\circ \tau_{X, TY}=\mu\circ T\tau_{X,Y}\circ \sigma_{TX, Y}$.
Now, suppose our monad isn't commutative. Is there a way to make it commutative in a canonical way?
As an example, could we obtain the free commutative monoid monad from the free monoid monad over $\mathtt{Set}$ in a canonical way?
Could we find a left-adjoint to the forgetful functor from the category of commutative monads over $(\mathcal C, \otimes, e, \alpha, \lambda, \rho, \gamma)$ to the category of strong monads over $(\mathcal C, \otimes, e, \alpha, \lambda, \rho, \gamma)$? Could the first be seen as a reflexive subcategory of the latter?