TITLE: What are the group objects in the category of finite sets and bijections, and its functor category?
QUESTION [8 upvotes]: An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such that $X\mapsto \operatorname{hom}\left(X,G\right)$ is a (contravariant) functor from $\mathcal{C}$ to $\text{Grp}$.

Group objects of the category $\text{Set}$ are groups in the usual sense. Similarly, group objects of the category $\text{FinSet}$ are finite groups.
Assuming that good (non-tautological) descriptions exist,

What are the group objects of the category $\text{FinBij}$, the category whose objects are finite sets, and whose morphisms are bijections?
What are the group objects of the functor category of $\text{FinBij}$, the category whose objects are functors of $\text{FinBij}$, and whose morphisms are natural transformations?

REPLY [11 votes]: There is no group object: since you only have bijections in your category, for all G there exists an X such that hom(x,G) is empty, so not a group in any way at all.