That is to say, the mathematical objects at
both higher and lower levels of the ∈-structure constitute examples of the same
mathematical notion of a “set”, so that one may consider “bijections of sets” between those sets without violating the axiom of foundation. In some sense, the notion of a species may be thought of as a natural extension of this observation.
That is to say,
the notion of a “species” allows one to consider, for instance, speciesisomorphisms between species-objects that occur at different levels of the
∈-structure of the set theory under consideration - i.e., roughly speaking,
to “simulate ∈-loops” - without violating the axiom of foundation.
以上