Lagrangian cobordism functor in microlocal sheaf theory II

For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are equivalent to sheaves at the negative end $Sh_{\Lambda_-}(M \times \mathbb{R})$ together with the data of local systems $Loc({L})$ by studying sheaf quantizations for general noncompact Lagrangians. Thus we interpret the Lagrangian cobordism functor between $Sh_{\Lambda_\pm}(M \times \mathbb{R})$ as a correspondence parametrized by $Loc({L})$. This enables one to consider generalizations to immersed Lagrangian cobordisms. We also prove that the Lagrangian cobordism functor is action decreasing and recover results on the lengths of embedded Lagrangian cobordisms. Finally, using the construction of Courte-Ekholm, we obtain a family of Legendrians with sheaf categories Morita equivalent to chains of based loop spaces of the Lagrangian fillings.