  We develop a new setting for the exponential principle in the context of
multisort species, where indecomposable objects are generated intrinsically
instead of being given in advance. Our approach uses the language of functors
and natural transformations (composition operators), and we show that, somewhat
surprisingly, a single axiom for the composition already suffices to guarantee
validity of the exponential formula. We provide various illustrations of our
theory, among which are applications to the enumeration of (semi-)magic
squares.
