An interesting example of the notion appears in the context of Joyal’s species of structures.
A species is a functor from the category of finite sets and bijections to , and the functors that are obtained by taking left Kan extensions of species along the embedding are called analytic functors. Now taking left Kan extensions along is pseudomonic, and this implies that the coefficients of an analytic functor are unique up to isomorphism.
I think that in a sense pseudomonic functors are precisely the functors for which it makes sense to say that is uniquely determined by up to isomorphism (although we do not really need faithfulnes for this, bijectivity on isos suffices).
Revised on October 26, 2010 18:20:47
by Urs Schreiber