Dear Jon,

for constructing the opposite of a fibration you do not need at all
that the chosen cleavage is split. It is easy to see that fixing the
cartesian arrow also fixes the vertical arrow.

If everybody is happy with this solution I am fine. It is in the same spirit
as chosen pullbacks or chosen finite limits.
The reason for my reluctance is that nobody would consider as a
natural notion a vector space with a chosen basis.
The virtue of strong choice is that then such additional structure can be
pulled out of the hat at demand.


Thomas