I refer to "Sheaves in Geometry and Logic", by S. MacLane.

Let **C** be a category. Dealing with a *subobject* of an object $D \in \text{Ob}_{\mathbf C}$, one defines an equivalence relation between morphisms towards *D*:

Two monomorphisms $f:A\to D$, $g:B\to D$ with a common codomain

Dare calledequivalentif there exists an isomorphism $h\colon A\to B$ such thatgh=f. AsbobjectofDis an equivalence class of monos towardsD. The collection Sub_{C}(D) of subobject ofDcarries a natural partial order [...]. Then Sub_{C}(D) isthe setof all subobjects ofDin the categoryC.

I can't figure out *why* Sub_{C}(*D*) is a set, rather than a proper class! Indeed, we are considering something like an qeuivalence relation on

$\displaystyle \coprod_{A\in \text{Ob}} \text{Hom}_{\bf C}(A,D)$

which is not a set, as soon as **C** isn't small.

So, how can I avoid the problem?

Cis "a contavariant functor betweenCandSets". I obviously thought about throwing everything in a suitable universe, but MacLane never mention the Grothendieck's universes, so I believe there is another way. – tetrapharmakon Jul 28 '10 at 10:55editing helppage don't work... :( ) – tetrapharmakon Jul 28 '10 at 10:56