Imagine a semisimple abelian category $\mathcal{C}$, for example representations of a finite group.
Take two (nonsimple) objects $X, Y$ that are subobjects of a common object $Z$, and decompose them into simples, say $X \cong X\_1 \oplus X\_2 \oplus \cdots \oplus X\_n$ and $Y \cong Y\_1 \oplus Y\_2 \oplus \cdots \oplus Y\_m$.
If some of the $X\_i$ and $Y\_j$ are isomorphic and are being mapped onto the same objects in $Z$, say $X\_{i\_k} \cong Y\_{j\_k}$ for $k \in \{1, \ldots K\}$, then $X$ and $Y$ have a common subobject $S$, namely $S := X\_{i\_1} \oplus X\_{i\_2} \oplus \cdots \oplus X\_{i\_K}$ such that $S \hookrightarrow X \hookrightarrow Z = S \hookrightarrow Y \hookrightarrow Z$. It is a maximal object with this property, all other common subobjects are subobjects of it.


This looks a lot like a universal property, but I can't figure out which one. What especially intrigues me that I'm looking not for any objects, but for subobjects. How do I encode that into the universal property?



What is the universal property of being the maximal common subobject of two objects in a semisimple category?
One can define the category of subobjects of $Z$, where the objects are subobjects of $Z$ and the morphisms have to commute with the monomorphisms of the subobjects.
In this category, $S$ is simply the product of $X$ and $Y$.


