Note: the answer to this question addresses my concerns (I think), but does not contain any references. Hence I believe my question is not a duplicate. This question is related (I think).
Question: To what extent can we extend the notion of equivalence relation, or of a relation in general, to classes from sets (i.e. including proper classes)?
If it is a meaningful restriction, I am interested primarily in the case when the class is the objects of a category.
A reference will suffice for an answer and would be preferred as well -- this seems like it must be a standard topic of set theory, but since I am completely unfamiliar with the set theory literature, I do not know where to begin to look for clarification about (equivalence) relations for classes.
Motivation: Consider a category $\mathscr{C}$. We get transitivity and reflexivity of the objects for free from the axioms of a category. To force symmetry I believe we can just restrict to the subcategory with the same objects but which contains only the invertible morphisms. This subcategory is a groupoid or setoid if I'm not mistaken. However, the definition on Wikipedia says that a groupoid has to be a small category, which is more restricitive than what I have in mind. If I'm not mistaken, what I am describing above is just classifying the objects in a category up to equivalence.
I think a precise answer depends the specific relationship between classes and sets, the details of which depend on how you formalize classes. Grothendieck universes are one way of formalizing classes, other ways of formalizing classes are described in Mike Shulman's Set Theory for Category Theory.
The key (at least in the usual approaches) is that sets are identified with certain kinds of classes, namely the small classes. In practice, this means that sets admit the same kinds of manipulations and constructions as classes, plus additional manipulations and constructions due to being special kind of classes.
Accordingly, often a notion formulated for sets turns out to rely only on the manipulations and constructions that are valid for both sets and classes, so it can be formulated for classes. Equivalence relations are certainly one example, as are categories and groupoids, (which is why we can have large categories and large groupoids).
A less trivial example is the Yoneda lemma: it is valid not just for set-valued presheaves on a locally small category, but for class-valued presheaves on any category; however the notion of a class-valued presheaf is subtler that of a set-valued presheaf: it is at this level of complexity that Grothendieck universes are a helpful formalism.