It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite sets. It is a symmetric monoidal category with tensor product $\coprod$ and unit $\emptyset$. It contains a commutative monoid $\{\star\}$, which seems to be the universal one: For every symmetric monoidal category $\mathcal{C}$ the assignment $F \mapsto F(\{\star\})$ provides an equivalences of categories

$\mathrm{Hom}_{\otimes}(\mathsf{FinSet},\mathcal{C}) \cong \mathrm{CMon}(\mathcal{C}).$

Here, $\mathrm{Hom}_{\otimes}$ denotes the category of strong symmetric monoidal functors (not assumed to be strict).

**Question 1.** This should be well-known, is there a reference in the literature?

**Question 2.** Is there any description of the *category* $\mathsf{FinSet}$ using generators and relations, i.e. an elementary description of $\mathrm{Hom}_{\mathrm{Cat}}(\mathsf{FinSet},-)$? Since every map of finite sets is a bijection followed by a monotonic map, or vice versa, I expect that we need face maps, degeneracies and transpositions as generators. What are the relations? Is this written down in the literature?

**Question 3.** An answer to question 2 will also describe presheaves on $\mathsf{FinSet}$, which are simplicial sets with a certain extra structure. Do they have a geometric interpretation and are these geometric objects used somewhere? In some sense, this corrects the failure of the join of simplicial sets to be commutative.

*Edit** after Eric's comment*. Ok, Q3 was already answered on MO. The objects are called *symmetric simplicial sets*. Relevant papers are Higher Fundamental Functors for Simplicial Sets by M. Grandis and *Toposes generated by codiscrete objects in combinatorial topology and functional analysis* by F. W. Lawvere and Left-determined model categories and universal
homotopy theories (Section 3) by J. Rosicky and W. Tholen.