Question on choosing subobjects consistently

In his Tohoku paper, Grothendieck asserted with no proof that in any category it is possible to choose subobjects for each object so that each monomorphism is isomorphic to a unique subobject of the codomain and in such a way that a subobject of a subobject of an object is also one of the chosen subobjects of the original objects. Maybe I am being dense, but I don't see how this is always possible. Does anyone on the list? I also don't see what possible value there is in making such a choice, but this doubtless was not clear in 1957. The translation (and revision) is coming along fine and I expect to release it within a month. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]