Is this an error on Wiki's "Exact sequence" page? 
$\textbf{Short exact sequence}$

  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.
