Let L_t be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle S^*M. Let Sh(M, L_t)$ be the differential graded (dg) derived category of constructible sheaves on M with singular support at infinity contained in L_t. We state a few results about when the Legendrian isotopy leaves the sheaf category invariant. The deformation of Legendrian and sheaves has been used in proving non-equivariant coherent-constructible correspondence for the toric variety, giving a different proof of recent result by Kuwagaki.