Lagrangian cobordism functor in microlocal sheaf theory

Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $Lambda_-$ to $Lambda_+$, we construct a functor $Phi_L^*: Sh^c_{Lambda_+}(M) rightarrow Sh^c_{Lambda_-}(M) otimes_{C_{-*}(Omega_*Lambda_-)} C_{-*}(Omega_*L)$ between sheaf categories of compact objects with singular support on $Lambda_pm$ and its adjoint on sheaf categories of proper objects, using Nadler-Shendes work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.