Here is what Grothendieck actually said (in our translation): "Thus a subobject of $A$ is not simply an object of \C, but an object $B$, together with a monomorphism $u:B\to A$ called the \emph{canonical injection} of $B$ into $A$. (Nonetheless, by abuse of language, we will often designate a subobject of $A$ by the name $B$ of the corresponding object of \C.) The containment relation defines an \emph{order} relation (not merely a preorder relation) on the class of subobjects of $A$. It follows from the above that the subobjects of $A$ that are contained in a subobject $B$ are identified with the subobjects of $B$, this correspondence respecting the natural order." It all depends on what he meant by "are identified with". Maybe I am making too much of this. He regularly uses "=" for isomorphism (which I have mostly changed in the translation). Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]