I am reading the homological algebra by Hilton and Stammbach (Chapter 3.1) where an extension of modules $(A,B)$ is defined to be the exact sequences $0\to B\to X \to A\to0$ by an (well-understood) equivalent relation. We want to turn it into a set-valued bi-functor. But my question is: is it guaranteed that the quotient object is indeed a set? Can it be a proper class?
E.g. when we relax the restriction $0\to B\to X$, then I do not think it is so true that all equivalent classes form a set.
Equivalence classes of extensions of $A$ by $B$ are in bijection with classes degree $1$ cocycles in a complex $\hom(P,B)$ where $P$ is a projective resolution of $A$, so they are a set.