  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 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.
