  We consider a fibered Lagrangian $L$ in a compact symplectic fibration with
small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks
with Lagrangian boundary from the base to the total space. In case $L$ is a
product, we use this machinery to give a formula for the leading order
potential and formulate an unobstructedness criteria for the $A_\infty$
algebra. We provide some explicit computations, one of which involves finding
an embedded 2n+k dimensional submanifold of Floer-non-trivial tori in a 2n+2k
dimensional fiber bundle.
