Let $\Delta_+$ denote the category of finite ordinal numbers with monotonic maps (the subscript indicates that $0$ is included, so this is the [augmented simplex category][1]). This has a monoidal structure (given by the sum), which is not symmetric. But we can make it symmetric in a universal way, see [here][2] for the general procedure. Let us denote this symmetric monoidal category by $(\Delta_+)_{\mathrm{sym}}$.
 
**Question.** What is a more "concrete" symmetric monoidal category which is equivalent to $(\Delta_+)_{\mathrm{sym}}$?

Notice that this is not the symmetric monoidal category $\mathcal{F}$ of finite sets. Whereas $(\Delta_+)_{\mathrm{sym}}$ classifies algebra objects in symmetric monoidal categories, $\mathcal{F}$ classifies commutative algebra objects in symmetric monoidal categories. Hence, there will be a strong symmetric monoidal functor $(\Delta_+)_{\mathrm{sym}} \to \mathcal{F}$, which is essentially surjective, but not fully faithful.


  [1]: https://ncatlab.org/nlab/show/simplex+category
  [2]: https://mathoverflow.net/questions/117533/free-symmetric-monoidal-category-on-a-monoidal-category