The authors deal with a kind of convex composite multiobjective nonsmooth programming. They present first order Lagrangian necessary conditions and new sufficient optimality conditions of efficient and properly efficient solutions, and duality results for (convex) composite problems, thus providing a unified framework for studying multiobjective optimization problems in connection with first order optimality and duality theory. The results generalize the corresponding results for convex and many classes of nonconvex problems, commonly encountered in the study of first order global optimality and duality theory. The authors present also some scalarization properties for the composite model problem under appropriate conditions.