The study of coalgebraic methods, as one of the active areas for theoretical computer science in recent years, is widely used in concurrence computing model, automat theory, and the foundations of object-oriented technology. Through using category theory, this paper investigates the properties of subcoalgebras, especially, the properties of subcoalgebras on Set, the category of sets and functions. This paper shows that all the subcoalgebras on Set are regular. Furthermore, using the correspondence between the cocongruence corelations and the subcoalgebras on Set, a way to construct the co-generated subcoalgebra is given in this paper.