We define a unital A(8)-category Fuk(R x Y ) whose objects are exact Lagrangian cobordisms in the symplectization of Y = P x R, with negative cylindrical ends over Legendri-ans equipped with augmentations. The morphism spaces hom(Fuk(RxY))(S-0, S-1) are given in terms of Floer complexes Cth(+)(S-0, S-1) which are versions of the Rabinowitz Floer complex defined by Symplectic Field Theory (SFT) techniques.