I have been poking around the internet and nlab looking at the concept of large and small categories. My original focus was locally presentable categories of categories and I was thinking of finding categories like Hilb and Group and Set as colimits over diagrams of compact objects. This lead to trying to define "largeness" and "smallness" in terms of colimits, ie, large categories as colimits of diagrams of small categories.

There is plenty of work done on internalizing the concept of "category". We even know that a "small category" is one that is internal to set. Is it possible to internalize how the concept of smallness is related to largeness? I had one thought on the matter, which was to have some ambient category that could host SET, and so all the small categories are internal there. We want to define largeness as maps from internal categories in other categories into this. That is a stupid attempt, but its all I have. Is it even possible to have a category which contains large categories and small categories?

I just don't see anywhere an attempt to internalize the concepts of how largeness relates to smallness.

Ensis the (superlarge) category of large sets, thenCat(Set)is somehow simultaneously the internal category of the categories internal to the internal categorySet, and also a subcategory ofCat(Ens), $\endgroup$