# intersections in abelian category

Let $\mathcal{A}$ be an abelian category. We fix an object $A$ and we consider the category $mono(A)$ whose objects are the monomorphisms $u:B\rightarrow A$ and where a morphism from $u:B\rightarrow A$ to $v:C\rightarrow A$ is a morphism $w: B\rightarrow C$ such that $v\circ w=u$. We can define the intersecion of $u$ and $v$ as a product of $u$ and $v$, that is, it is a monomorphism $w:K\rightarrow A$ with morphisms $\pi_1:K\rightarrow B$ and $\pi_2:K\rightarrow C$ such that for any other monomorphism $w':K'\rightarrow A$ with morphisms $p_1:K'\rightarrow B$ and $p_2:K'\rightarrow C$ there exists a unique map $p:K'\rightarrow K$ such that $\pi_i\circ p=p_i$.

Is this the correct definition of intersections of subobjects in an abelian category? and if it is, which are the general conditions that we have to impose on $\mathcal{A}$ to ensure that every pair of subobjects of any object have intersection?

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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)?
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!).