Enriched Categories of Coalgebras

Let V be a monoidal category with underlying ordinary category V 0 .Denote by A an enriched category over V.For any V-endofunctor T : A → A, a T -coalgebra is defined as a pair (A, τ A ) consisting of an object A in obA together with a V 0 -morphism τ A : I → A(A, T A), where I is the unit.The object A is called the underlying object of the T -coalgebra (A, τ A ) and τ A is called its coalgebra structure.Assuming that V is symmetric and admits equalizers, it is shown that T -coalgebras form a V-category, denoted by A T .Consequently, the correspondence U T : obA T → obA that assigns each T -coalgebra (A, τ A ) to its underlying object A is a V-functor.Moreover, if V is closed and complete, and A is a small V-category, then the underlying V-functor U T : A T → A is shown to create weighted colimits and weighted limits that are preserved by T .