Finite Sets Category

Clearly, finite sets and functions between them form a category $\mathsf{FinSet}$, which is a full subcategory of $\mathsf{Set}$. $\mathsf{FinSet}$ is a symmetric monoidal category with disjoint union as the monoidal product and the empty set as the monoidal unit. Moreover, $\mathsf{FinSet}$ is a symmetric monoidal category with the braiding given by the natural isomorphism between $A\bigsqcup B$ and $B\bigsqcup A$.
Any two sets of the same cardinality are isomorphic in $\mathsf{FinSet}$, so we can define the category of finite cardinals $\mathsf{FinCard}$ as the skeleton of $\mathsf{FinSet}$. The objects of $\mathsf{FinCard}$ are the finite cardinals $0,1,2,\dots$, and morphisms are functions between them.
On the other hand, the category of finite posets $\mathsf{FinPos}$ is also a subcategory of $\mathsf{FinSet}$, and its skeleton is the category of finite ordinals $\mathsf{FinOrd}$, whose objects are the finite ordinals $0,1,2,\dots$ and morphisms are order-preserving functions between them. $\mathsf{FinOrd}$ is also called the simplex category $\mathsf{\Delta }$.

We can describe $\mathsf{FinCard}$ and $\mathsf{FinOrd}$ using graphs, where each object $n$ is represented by $n$ dots (i.e., vertices), and morphisms are represented by strands (i.e., directed edges) between dots. In $\mathsf{FinCard}$, we only require that for any dot in the domain, there is only one strand coming out; while in $\mathsf{FinOrd}$, the strands must also not cross each other. For example, a morphism $f:4\to 3$ in $\mathsf{FinOrd}$ can be represented as follows:

Proposition

$\mathsf{\Delta }$ is generated by

under the relations

Proposition

Any morphism $f$ in $\mathsf{FinCard}$ is a composition of a bijection $g$ (i.e., permutation) and an order-preserving function $h$: $f=h\circ g$.

Remark

Note that the order matters here. In general, we cannot write $f=g\circ h$.

Corollary

$\mathsf{FinCard}$ is generated by the following morphisms:

Theorem

$(\mathsf{FinOrd},+,0)$ is the free monoidal category on a single monoid object $1$. That is, for any monoidal category $(\mathsf{C},\otimes ,1)$ and any monoid object $M$ in $\mathsf{C}$, there is a unique monoidal functor $F:(\mathsf{FinOrd},+,0)\to (\mathsf{C},\otimes ,1)$ such that $F(1)=M$.
In other words, there is a canonical equivalence of categories
$\mathsf{MonFunc}(\mathsf{FinOrd},\mathsf{C})\cong \mathsf{Mon}(\mathsf{C}),$
where $\mathsf{MonFunc}(\mathsf{FinOrd},\mathsf{C})$ is the category of monoidal functors from $\mathsf{FinOrd}$ to $\mathsf{C}$, and $\mathsf{Mon}(\mathsf{C})$ is the category of monoid objects in $\mathsf{C}$.