Adding to Steve's most appropriate comments. One also wants to have the so-called Beck-Chevalley conditions. Recall that a fibration has internal sums iff it is a bifibration where cocartesian arrows are stable under pullbacks along cartesian arrows. This property is easily seen to be preserved by composition. But P has internal products iff P^op has internal sums. However, we don't have (P \circ Q)^op = P^op \circ Q^op in particular because since the right composite doesn't exist. (If P is a fibration over BB then P^op still is a fibration over BB and not over BB^op). Still I guess one can check directly that composition preserves the property of having small products. Maybe it's even in Bart Jacob's book? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]