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This paper develops a distributed solution to the fully-heterogeneous containment control problem (CCP), for which not only the followers' dynamics but also the leaders' dynamics are non-identical. A novel formulation of the fully-heterogeneous CCP is first presented in which each follower constructs its virtual exo-system. To build these virtual exo-systems by followers, a novel distributed algorithm is developed to calculate the so-called normalized level of influences (NLIs) of all leaders on each follower and a novel adaptive distributed observer is designed to estimate the dynamics and states of all leaders that have an influence on each follower. Then, a distributed control protocol is proposed based on the cooperative output regulation framework, utilizing this virtual exo-system. Based on estimations of leaders' dynamics and states and NLIs of leaders on each follower, the solutions of the so-called linear regulator equations are calculated in a distributed manner, and consequently, a distributed control protocol is designed for solving the output containment problem. Finally, theoretical results are verified by performing numerical simulations.