This work studies the discrete-time controllability of a composite network formed by factor networks via Cartesian products. Based on the Popov-Belevitch-Hautus test and properties of Cartesian products, we derive the algebra-theoretic necessary and sufficient conditions for the controllability of the Cartesian product network (CPN), which is devoted to carry out a comprehensive study of the intricate interplay between the node-system dynamics, network topology and the controllability of the CPN, especially the intrinsic connection between the CPN and its factors. This helps us enrich and perfect the theoretical framework of controllability of complex networks, and gives new insight into designing a valid control scheme for larger-scale composite networks.