Completion of categories

Abstract

Let be functors. We know that the class [F, G] of all functorial morphisms from F into G is not always a set, so that we can not speak in general about the category of all functors from into et. But instead of all functors from into we can consider only so-called proper functors, which were defined by Isbell [3]. In this way we can define a category , the category of all proper functors from into . This category contains as a full subcategory and it coincides with the category in the case when is a small category.