In this paper, we study the one-to-one correspondence between the dual NLS hierarchy and derivative NLS hierarchy, and associated Hamiltonian conservation laws. The first part, we give the correspondence between these two recursion operators according to the gauge transformation between the dual NLS and derivative NLS equations. And base on these relationships, we prove that these two equation hierarchies have the same gauge transformation. The sencond part, we propose the correspondence between these two equation hierarchies about the variation derivative of Hamiltonian functional. And then, we obtain to the relationship about their associated Hamiltonian conservation laws.