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Consider sheaves on manifolds with microsupport on a (singular) Legendrian subset inside the cosphere bundle, which are in particular (real) constructible sheaves. By microlocalization, one can define a sheaf of categories on the singular Legendrian in the cophere bundle, called microsheaves. We will study duality and exact sequences arising from the pair of categories. More precisely, we show that the microlocalization functor from sheaves to microsheaves together with its left adjoint admits a strong smooth relative Calabi-Yau structure. This is a non-commutative analogue of the orientation class which induces the Poincare-Lefschetz duality on manifolds with boundary. Under some extra assumptions, we can show that the adjunction is a spherical adjunction and the inverse dualizing bimodule is the Serre functor. We will explain the connections to Fukaya categories, Legendrian contact homologies and certain cluster varieties. This is joint work in preparation with Chris Kuo.