Duality and kernels in microlocal geometry

We study the dualizability of sheaves on manifolds with isotropic singular supports ${\mathrm{Sh}}_{\mathrm{\Lambda }}(M)$ and microsheaves with isotropic supports ${\mathrm{\mu sh}}_{\mathrm{\Lambda }}(\mathrm{\Lambda })$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports ${\mathrm{Sh}}_{\mathrm{\Lambda }}^b(M{)}_0$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects ${\mathrm{Sh}}_{\mathrm{\Lambda }}^c(M{)}_0$ when the wrap-once functor is an equivalence, for instance, when $\mathrm{\Lambda }$ is a full Legendrian stop or a swappable Legendrian stop.