On Satellites in Arbitrary Categories

We generalize the definition of satellites with respect to presheaves (and copresheaves) with trace in the sense of [1]; 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. We shall denote by S(X,Y ), where X and Y are categories, the category of which the objects are graphs S together with two diagrams F = F (S) : S −→ X and G = G(S) : S −→ Y ; the morphisms in S(X,Y ) are defined in the natural way. We have an isomorphism S(X,Y ) ≈ S(Y , X) (1)