$\textbf{Short exact sequence}$
The most common type of exact sequence is the '''short exact sequence'''. This is an exact sequence of the form $A \;\overset{f}{\hookrightarrow}\; B \;\overset{g}{\twoheadrightarrow}\; C$ where $f$ is a monomorphism and $g$ is an epimorphism. In this case, $A$ is a subobject of $B$, and the corresponding quotient is isomorphic to $C$: $C \cong B/f(A)$ (where $f(A)$ = im($f$)).
Now, does this make sense to say that $A$ is a subobject of $B$ rather than $f(A)$ is a subobject of $B$? My understanding of category theory is that you can play madlibs with it and get a coherent statement in other environments.
If we're talking about modules, for example, then I would say it's fine to call $A$ a subobject of $B$. As $f$ is injective $A$ is isomorphic to $f(A)$ and isomorphisms are commonly repressed when speaking about algebraic objects.
On the other hand, if we're being completely abstract then a subobject of $B$ is an equivalence class of monomorphisms into $B$. In this case neither $A$ nor $f(A)$ is technically a subobject of $B$, the maps $A \to B$ and $f(A) \to B$ are representative elements for a subobject, and in this case they represent the same subobject.