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

A-infinity category of Lagrangian cobordisms in the symplectization of PxR