  We construct projective limit of projective sequence in the following
categories: Archimedean order unit spaces with unital positive maps and
operator systems with unital completely positive maps. We prove that inductive
limit and projective limit in these categories are in duality, provided that
the dual objects remain in the same categories and the maps are order
embeddings. We generalize a result of Choi and Effros on matrix-ordered duals
of finite-dimensional operator systems to separable case.
