The definition of a (combinatorial) species $F$ emphasizes that any bijection $\phi\, :\, A \to B$ between support sets $A$ and $B$ must induce a bijection between $F[A]$ and $F[B]$ (the $F$-structures on $A$ and $B$ respectively), referred to as "transport of $F$-structures along $\phi$" in the standard textbooks by Bergeron, Labelle, and Leroux. For all the usual examples of species, the induced bijection between $F[A]$ and $F[B]$ is the "obvious" one. I trace this to the fact that for all these species, every structure on a support set $A$ is (or can be viewed as) a set in which each element is either (1) an element of $A$ or (2) a set in which each element is either (1) an element of $A$ or (2) a set in which each element is either ... (continued to a finite level of recursion).

If the notion of a species was restricted to structures formed in this way, the emphasis on induced bijections would be unnecessary---an induced bijection could always be found by letting $\phi$ "pass thru" the enclosing braces of all the sets.

The question is: would there be any significant loss of generality (as opposed to a loss of conceptual simplicity) in doing so?

More specifically, is there an example of a species whose structures cannot be expressed as sets of the above type?