Union in an abelian category

Question

I'm working on the following exercise. I have most of it, but there's one detail at the end that I can't work out.

Let $\{A_i\}$ be a family of subobjects of an object $A$. Show that if $\mathcal{A}$ is abelian and cocomplete then there is a smallest subobject $\sum A_i$ of $A$ containing all the $A_i$.

Most of it is worked out on this question, but I'm confused about how to apply the universal property of the image.

Specifically, $A_i\hookrightarrow A$ are monomorphisms and there's an induced map $\coprod A_j\rightarrow A$. The object $\sum A_j$ is simply the image of this map. Suppose that $Y$ is a subobject of $A$ that contains all the $A_i$. Then there are monomorphisms $A_i\hookrightarrow Y\hookrightarrow A$ (and moreover this composition must be the original $A_i\hookrightarrow A$ since subobjects are equivalence classes of monics with the same domain), so there's an induced map by universal property of the coproduct $\coprod A_j\rightarrow Y$.

Now, in order to apply the universal property of the image like the answer suggests, I need for $\coprod A_j\rightarrow \sum A_j\rightarrow A$ to be the same factorization as $\coprod A_j\rightarrow Y\rightarrow A$. This is where I get stuck. I only get that $A_i\rightarrow \coprod A_j\rightarrow \sum A_j\rightarrow A$ is the same factorization as $A_i\rightarrow\coprod A_j\rightarrow Y\rightarrow A$, and $A_i\rightarrow \coprod A_j$ is in general not an epimorphism. Since I can't cancel on the right, I don't know how to satisfy the requirements to use U.P. for image.

Answer 1

This is immediate from the definition of a coproduct. By definition, for any collection of maps $f_i:A_i\to A$, there is a unique map $f:\coprod A_j\to A$ such that $fs_i=f_i$ for all $i$, where $s_i:A_i\to \coprod A_j$ is the inclusion. The uniqueness part of this means that if two maps $\coprod A_j\to A$ have the same compositions with $A_i\to\coprod A_j$ for each $i$, then the two maps are equal.

Answer 2

Here's a slightly different approach. I haven't checked this in detail though.

Let $(A_i)_{i\in I}$ be your family of subobjects of $A$. For a finite $I_0\subseteq I$ then $\sum_{i\in I_0}A_i$ is also a subobject of $A$. Now consider the collection of all these finite sums; these form a directed system and so have a colimit. Is the image of this colimit in $A$ the required subobject of $A$?