In this paper, the concepts and the properties of conjugate maps and directional derivatives of convex vector functions from a subset of $R^n$ to $R^m$ with respect to a convex, closed and pointed cone are presented on the base of the notions of Pareto-supremum and Pareto-infimum. Some well-known results in the scalar case are generalized to the vector case. An application of conjugate maps to the dual problem is shown.