Free monoidal category over a set From nLab's article on coherence theorems, there seems to be a notion of free monoidal category over a set $S$.
I guess this corresponds to the left adjoint to the functor $Ob : MonCat \to Set$ which given a monoidal category $C$, it gives back the set of objects of $C$. The free monoidal category over $S$ would be a monoidal category $C$ together with a function $i : S \to Ob C$ such that for any other monoidal category $D$ and function $f : S \to Ob D$, I get a unique monoidal functor $\bar{f} : C \to D$ such that $Ob \bar{f} \circ i = f$.
After thinking of that definition, I thought that the objects of the free monoidal category over $\{ * \}$ would be $I$, $\{ * \}$, $\{ * \}^2$, etc.
Then, I thought that a similar presentation can be made for a free strict monoidal category over a set $S$, by replacing "monoidal" by "strict monoidal". 
However, I started to wonder, wouldn't the free strict monoidal category over $\{ * \}$ satisfy the condition of being the free (non-necessarily strict) monoidal category over $\{ * \}$? If that was the case, I think the nLab page wouldn't make much sense. What's wrong here? Is my definition of free monoidal category?
 A: The construction you describe is exactly similar to the free-strict-monoidal category but it is not the free-monoidal category: the difference being that in a strict monoidal category associators and left and right units are required to be the identities of the respective objects, a requirement that you did not add in your construction.
The free-monoidal-(biased)-category $F(S)$ over $S$ should have:


*

*as set of object the set of all the formal expressions you can build from the set of symbols in $S$ with the operators $\otimes$ and $1$

*as set of morphisms is generated by associators, left and right unit (and their inverses) through application of compositions and monoidal products, quotienting by the various equations needed for a monoidal categories (associativity and identities for compositions, functoriality of the monoidal product and most importantly coherence conditions).


One of the interesting things is that coherence conditions give you the ability to describe more explicitly the set of morphisms in this category:
for each pair of objects in $F(S)$ you have exactly one morphism between two objects if and only if they are equivalent up to associativity and unit.
This data give you a category with the wished property: namely that for every monoidal category $\mathbf C$ and a function $f \colon S \to Ob(\mathbf C)$ there is a monoidal functor $\bar f \colon F(S) \to \mathbf C$ such that the equation $f=Ob(\bar f) \circ i$ holds, where $i \colon S \to Ob(F(S))$ is the obvious embedding sending every element of $S$ in itself seen as an element of $F(S)$.
P.s.: I realize that what I've described is not a fully formal description of what the free monoidal category over the set $S$ should be, but I hope that is enough to get the idea.
P.p.s.: above I've called $F(S)$ to distinguish it from the free-unbiased-monoidal categories described in Leinster's book Higher Operads, Higher categories.
