It is mentioned in the introduction to  that (Cartesian) differential categories might be the unifying framework for differentiation in various branches of mathematics including combinatorics. It is also mentioned in  and other papers on tangent categories that there are tangent categories of combinatorial species. I don't see how the obvious definition of the category of species can be made into a tangent category, but maybe this is true for some other category of species.
Questions: What is the relationship between combinatorial species and differential/tangent categories? Is there a differential or tangent category of species? Is the operation of differentiation of species related to these structures?
 Blute, R.; Cockett, J. R. B.; Seely, R. A. G., Cartesian differential categories, Theory Appl. Categ. 22, 622-672 (2009).