In a previous question, a description of the free monoidal category over a set was given. Basically, it consists of formal expressions as objects and morphisms generated by associators and unitors, quotiened by the equations of monoidal categories.

When one says that this or that is the free monoidal category over a set $S$, I guess one means that this construction is the left adjoint to the forgetful functor (just gives back the objects) $\mathrm{MonCat} \to \mathrm{Set}$, where $\mathrm{MonCat}$ consits of monoidal categories as objects and monoidal functors of some kind as morphisms. What I do not know is: what is this some kind? Strict monoidal functors? Strong monoidal functors? Lax monoidal functors?

It is usally stated that the functor from the universal property is strict, so it seems that the first one is the correct option. If so, what would be the free monoidal category over a set with respect to the functor $\mathrm{MonCat}_l \to \mathrm{Set}$, where $\mathrm{MonCat}_l$ the category of monoidal categories and lax monoidal functors between them? What about the one from $\mathrm{MonCat}_{st}$ (strong monoidal functors)?


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