Automata and Coalgebras in Categories of Species

We study generalized automata (in the sense of Adámek-Trnková) in Joyal’s category of (set-valued) combinatorial species, and as an important preliminary step, we study coalgebras for its derivative endofunctor \(\partial \) and for the ‘Euler homogeneity operator’ \(L\circ \partial \) arising from the adjunction \(L\dashv \partial \dashv R\) .