  We define a unital $A_\infty$-category whose objects are exact Lagrangian
cobordisms in the symplectization of $Y=P\times\mathbb{R}$, with negative
cylindrical ends over Legendrians equipped with augmentations. The morphism
spaces are given in terms of Floer complexes $Cth_+(\Sigma_0,\Sigma_1)$ which
are versions of the Rabinowitz Floer complex defined by Symplectic Field Theory
(SFT) techniques.
