The smallest subobject $\sum{A_i}$ containing a family of subobjects {$A_i$} In an Abelian category $\mathcal{A}$, let {$A_i$} be a family of subobjects of an object $A$. How to show that if $\mathcal{A}$ is cocomplete(i.e. the coproduct always exists in $\mathcal{A}$), then there is a smallest subobject $\sum{A_i}$ of $A$ containing all of $A_i$?
Surely this $\sum{A_i}$ cannot be the coproduct of {$A_i$}, but I have no clue what it should be.

You are quite right that it can't be the coproduct, since that is in general not a subobject of $A$. Here are two ways of constructing the desired subobject:


*

*As Pierre-Yves suggested in the comments, the easiest way is to take the image of the canonical map $\bigoplus_i A_i \to A$. This works in any cocomplete category with unique epi-mono factorisation.

*Alternatively, the subobject $\sum A_i$ can be constructed by taking the colimit over the semilattice of the $A_i$ and their intersections. This construction can be carried out in any bicomplete category, but is not guaranteed to give a subobject of $A$ unless the category is nice enough.
