intersections in abelian category

How are you defining subobject? Do you declare a monomorphism $B \to A$ to be a subobject of $A$ (like Mitchell), or do you define subobjects of $A$ to be certain equivalence classes of monics with target $A$ (like Mac Lane or Freyd)?

In any case, you have the right idea.

If we take the "subobjects are equivalence classes" definition, then recall that the class of subobjects of a given object has a natural partial order. If $u: B \to A$ and $v: C \to A$ represent two subobjects of $A$, we declare $u \leq v$ iff $u$ factors through $v$. The intersection of two subobjects is then their greatest lower bound with respect to this order (as it should be!).

Now, given an arbitrary abelian category, the intersection of any pair of subobjects (of a fixed object) always exists. This is Theorem 2.13 in Freyd's book (ftp://ftp.sam.math.ethz.ch/EMIS/journals/TAC/reprints/articles/3/tr3.pdf), and it's not a hard proof (it's the third theorem he proves about abelian categories).