I would like to know if you know of any literature that has a good discussion of this kind.

Thank you for your answer! It resolves some of my concerns. I have a question. As you said, almost all interesting categories are essentially small in the context of Grothendieck groups. Thus I want to construct K_0 as a functor from the category of essentially U-small abelian categories to the category of U-small abelian groups. However, I think that we cannot consider the set of all U-small abelian groups. Is it natural to fix a universe V containing U and consider the category of U-small abelian groups belonging to V?

As I write at the end of the text, I only consider categories such that ObC and MorC are sets. For example, we consider the category of abelian groups belonging to $\mathbb{U}$ instead of the category of all abelian groups. In such a case, I think that the problem which you described would not arise. I am a beginner in these matters, so I apologise if I have made any major mistakes.

@SimonHenry I would like to know why we should bother to prove whether a set is U-small in various situations.

My paper may be helpful for you. In this paper, I give a sufficient condition under which a given periodic triangulated category is equivalent to the periodic derived category of an algebra and study some periodic triangulated categories. In section 3, I give a detailed treatment of periodic derived categories.