On the origin of (combinatorial) species: a category-theoretical (de)construction of the theory of Joyal species

We give a brief overview of alternative theories of combinatorial species other than that given by Joyal [1]. Then, we construct a general theory of species which extends the concept of species for any two categories with definite categories as objects, and invertible morphisms (functors) in the first category, and this construct is equivalent to that of all the existing species defined on categories as the aforementioned, so that ordinary species, all types of weighted species, q-species [5], Mobius species [4], and L-species are just special cases. We adapt to this new scenario the classic constructs of assemblies of species, derivation and pointing. An analogue for exponentiation is constructed. Adapting the concept of arithmetic product yields a new type of function called "rational Dirichlet series".