There is a well known theorem from group theory, often called one of the "isomorphism theorems". If $S$ is a subgroup of $G$ and $N$ a normal subgroup of $G$ then $SN/N = S/(S \cap N)$.
If we let $\textbf C$ be a suitable category then for subobjects $A$ and $B$ of an object $C$ we can make sense of $A \cap B$, it is the fibered product of $A$ and $B$ over $C$. We can also make sense of $AB$, it is the smallest subobject of $C$ which contains both $A$ and $B$. I.e the join of $A$ and $B$ in the poset of subobjects of $C$.
If $\textbf C$ has terminal objects and $m: Y \rightarrow X$ is monic we can define $X/Y$ as the pushout of $* \leftarrow Y \xrightarrow m X$.
Then my question is this. In a suitable category $\textbf C$ is it true that $AB/B = A/(A \cap B)$ for subobjects $A$ and $B$ of an object $C$?