Functors from Lagrangian correspondences for wrapped sheaf categories

We consider the category of sheaves on a manifold with microsupport in a singular conic Lagrangian, and more generally microsheaves on the Lagrangian skeleton of a Weinstein manifold with stop, which are equivalent to ind-completions of partially wrapped Fukaya categories. Compact objects in these categories are obtained as wrappings of constructible sheaves with perfect stalks; they correspond to objects built from exact Lagrangians in finitely many steps. We give a microlocal criterion on constructible (micro)sheaves with perfect (micro)stalks which guarantees that the corresponding wrapped sheaves induce integral kernels that preserve compact objects of these categories. This generalizes a result of Gao on Lagrangian correspondences between fully wrapped Fukaya categories.