[Diagram depicting a pullback of spans.]
It is clear that composition is well-defined and associative. Thus when the isomorphism classes of diagrams (3) form a set (e.g. if every object of M has a set of subobjects) then QM is a well-defined category. We assume this to be the case from now on.
Back in the good old days when people cared about "size issues"...
Source: Quillen, Higher algebraic K-theory: I












