Is there a left adjoint to the forgetful functor from cartesian monoidal categories to symmetric monoidal categories? And if so: how can it be described explicitely.

In more detail: Given a symmetric monoidal category $\mathbb V$ i am looking for a monoidal functor

$$\chi:\mathbb V \to \mathcal C(\mathbb V)$$

into a cartesian monoidal category $\mathcal C(\mathbb V)$ such that every other monoidal functor

$$\varphi:\mathbb V \to \mathbb C $$

into a cartesian monoidal category $\mathbb C$ factors 'uniquely' over $\chi$.