Generalizing results of Herrlich and Nel, the author characterizes by means of smallest proper structures those objects X of an initially structured category for which X x—has a right adjoint, and describes the corresponding function spaces. It is shown that reduction to finally and initially dense classes is possible. The results are applied to epireflective subcategories of the category of limit spaces containing a finite non-indiscrete space, in particular to epireflective subcategories of TOP.