It is a classical theorem of Freyd that if a small category is complete (has all small limits—in fact, having small products suffices), then it is a preorder (has at most one morphism between any two objects). The proof of this theorem (which can be found here or in CWM) is non-constructive, i.e. it uses the Law of Excluded Middle. Therefore, it can potentially fail in the internal logic of an elementary topos. And in fact, it does fail in the effective topos, and more generally in realizability topoi, where there do exist small complete categories that are not preorders.
However, I have heard it said that Freyd's theorem cannot fail in a Grothendieck topos; i.e. that a small complete category in a Grothendieck topos must still be a preorder—despite the fact that the internal logic is still in general intuitionistic, so that Freyd's proof cannot work. Can someone explain why this is, or (even better) give a reference containing a proof?