On satellites in arbitrary categories
  We generalize the definition of satellites with respect to presheaves (and copresheaves) with trace in the sense of Inassaridze; a presheaf with trace is replaced by a graph with a pair of diagrams defined on it.   We show that the right satellite functor is left adjoint to the left satellite functor, and that a functor having a right (left) adjoint preserves right (left) satellites.   In particular cases the construction of satellites is given. 
George Janelidze