  We continue investigations of reasonable ultrafilters on uncountable
cardinals defined in math.LO/0407498. We introduce stronger properties of
ultrafilters and we show that those properties may be handled in lambda-support
iterations of reasonably bounding forcing notions. We use this to show that
consistently there are reasonable ultrafilters on an inaccessible cardinal
lambda with generating system of size less than 2^lambda . We also show how
reasonable ultrafilters can be killed by forcing notions which have enough
reasonable completeness to be iterated with lambda-supports (and we show the
appropriate preservation theorem).
