In Andreas Blass's famous paper `Seven Trees in One', the existence of a natural bijection between binary trees and 7-tuples of binary trees is related to the equation $T^7 = T$ being satisfied by a complex number which satisfies the equation $T = 1 + T^2$ (the categorification of which is satisfied by the set of binary trees).


In a similar vein, Leinster states that there are (for a non-standard definition of cardinality) $e$ finite sets up to isomorphism. Specifically, for each finite cardinality $k$, there is one way to specify a list of $k$ distinct elements, but this counts each set $k!$ times (one for each permutation), so there are $\dfrac{1}{k!}$ sets of size $k$ and $\dfrac{1}{0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \cdots = e$ finite sets.


<http://quomodocumque.wordpress.com/2008/11/26/tom-leinster-on-entropy-diversity-and-cardinality/>


Now, this result can be generalised. Consider a set of $k$ elements, each of which is coloured with one of $m$ non-equivalent colours. Up to isomorphism, there are now $\dfrac{m^k}{k!}$ $m$-coloured sets of size $k$, so $e^m$ finite $m$-coloured sets up to isomorphism.


Let $S\_m$ be the collection of finite sets (for concreteness, subsets of $\mathbb{N}$) whose elements are $m$-coloured. Note that $e^m e^n \equiv e^{m+n}$, which suggests the following categorification:


"There exists a natural bijection between $S\_m \times S\_n$ and $S\_{m+n}$."


Are there any prior results in this area? In particular, is there a natural bijection? (Such a bijection should generalise to when $\mathbb{N}$ is replaced by any infinite set, and should be constructive.)



Natural bijection between sets with coloured elements?
If I take an $m$-coloured set and an $n$-coloured set, their disjoint union (using disjoint sets of colours) is an $(m+n)$-coloured set. Conversely, if I have an $(m+n)$-coloured set, it has a subset which uses the first $m$ colours, and another subset which uses the other $n$...


While this answer is (perhaps unexpectedly) short, it is worth mentioning that there is a fairly good theory of phenomena like this: Joyal's theory of "combinatorial species". There's a [survey article](http://bergeron.math.uqam.ca/files/2013/11/book.pdf) online nowadays.


