We slightly improve on characterization results already in the literature for base revision. We show that in order to axiomatically characterize revision operators in a logic the only conditions this logic is required to satisfy are: finitarity and monotonicity. A characterization of limiting cases of revision operators, full meet and maxichoice, is also offered. In the second part of the paper, as a particular case, we focus on the class of graded fuzzy logics and distinguish two types of bases, naturally arising in that context, exhibiting different behavior.