A finite dimensional analogue of the story of  Floer homology for 3 manifold with boundary is a construction which gives a filtered A infinity functor from Lagrangian correspondence. In the first part I will explain such a construction. To associate a filtered A infinity category to a symplectic manifold we need to take a finite set of Lagrangian submanifold and also we need a certain transversality assumption on the Lagrangian submanifolds for the construction of a filtered A infinity functor. In the second part I will explain the process to take a completion of filtered A infinity functor which could be used to resolve the problem. The completion I will discuss is related to an enumerative aspect of Lagrangian Floer theory such as spectral invariant or torsion exponent.