  To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category,
the augmentation category. An exact Lagrangian cobordism between two Legendrian
knots gives a functor of the augmentation categories of the two knots. We study
the functor and establish a long exact sequence relating the corresponding
cohomology of morphisms of the two ends. As applications, we prove that the
functor between augmentation categories is injective on the level of
equivalence classes of objects and find new obstructions to the existence of
exact Lagrangian cobordisms in terms of linearized contact homology and ruling
polynomials.
