Lagrangian cobordism functor in microlocal sheaf theory I

Let ${\Lambda }_{\pm }$ be Legendrian submanifolds in the cosphere bundle $T^{\ast ,\infty }M$. Given a Lagrangian cobordism $L$ of Legendrians from ${\Lambda }_{-}$ to ${\Lambda }_{+}$, we construct a functor ${\Phi }_L^{*}:{\mathrm{Sh}}_{{\Lambda }_{+}}^c\left(M\right)\to {\mathrm{Sh}}_{{\Lambda }_{-}}^c\left(M\right){\otimes }_{C_{-*}\left({\Omega }_{*}{\Lambda }_{-}\right)}{C}_{-*}\left({\Omega }_{*}L\right)$ between sheaf categories of compact objects with singular support on ${\Lambda }_{\pm }$ and its right adjoint on sheaf categories of proper objects, using Nadler–Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high-dimensional Legendrian submanifolds.