It is well known that the number $D_n$ of derangements (fixed point free permutations) on a set of $n$ elements is exactly $\left[\dfrac{n!}e\right]$, the closest integer number to $\dfrac{n!}e$.
Using groupoid cardinality, one assigns exactly $e=\sum\limits_n\frac1{n!}$ as the cardinality of the (skeleton of the) category of finite sets.
Is there any deeper heurusistic (or even a clear proof) of the above formula for the number of derangements in the context of groupoid cardinalities?