Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q \cap N = 0$.
I've got just a short question what maximality means here: Does it mean that every other $Q'$ with $Q' \cap N = 0$ is contained in $Q$, or does it mean that every $Q'$ with $Q' \cap N = 0$ and $Q \subseteq Q'$ satisfies $Q = Q'$? Usually the latter is meant by maximal, right? However, then I don't understand at all the proof (sketches) in the section on essential monomorphisms and injective hulls in Gabriel's thesis.