  We study the dualizability of sheaves on manifolds with isotropic singular
supports $\operatorname{Sh}_\Lambda(M)$ and microsheaves with isotropic
supports $\operatorname{\mu sh}_\Lambda(\Lambda)$ and obtain a classification
result of colimit-preserving functors by convolutions of sheaf kernels.
Moreover, for sheaves with isotropic singular supports and compact supports
$\operatorname{Sh}_\Lambda^b(M)_0$, 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 $\operatorname{Sh}_\Lambda^c(M)_0$
when the wrap-once functor is an equivalence, for instance, when $\Lambda$ is a
full Legendrian stop or a swappable Legendrian stop.
