This study addresses the adaptive control and function projective synchronization problems between 2D Rulkov discrete-time system and Network discrete-time system.Based on backstepping design with three controllers, a systematic, concrete and automatic scheme is developed to investigate the function projective synchronization of discrete-time chaotic systems.In addition, the adaptive control function is applied to achieve the state synchronization of two discrete-time systems.Numerical results demonstrate the effectiveness of the proposed control scheme.