  Using quilted Floer cohomology and relative quilt invariants, we define a
composition functor for categories of Lagrangian correspondences in monotone
and exact symplectic Floer theory. We show that this functor agrees with
geometric composition in the case that the composition is smooth and embedded.
As a consequence we obtain 'categorification commutes with composition' for
Lagrangian correspondences.

Symplectic Geometry;Differential Geometry