Functoriality for Lagrangian correspondences in Floer theory

We associate to every monotone Lagrangian correspondence a functor between Donaldson–Fukaya categories. The composition of such functors agrees with the functor associated to the geometric composition of the correspondences, if the latter is embedded. That is “categorification commutes with composition” for Lagrangian correspondences. This construction fits into a symplectic 2-category with a categorification 2-functor, in which all correspondences are composable, and embedded geometric composition is isomorphic to the actual composition. As a consequence, any functor from a bordism category to the symplectic category gives rise to a category valued topological field theory.