Let $MonCat$ denote the 2-category of monoidal categories and strict monoidal functors. Let $Cat$ denote the category of 2-categories. There is a forgetful functor $Forget:MonCat\rightarrow Cat$. Does this have a left adjoint?
Short answer: yes.
A little longer answer: you can apply theorem 1 in the following link, using the fact that $MonCat$ is the category of monoids in $Cat$, that $Cat$ is monoidal (with respect to the cartesian product), it has countable coproducts and that products distributes with coproducts.
Hope this helps.