TITLE: If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic?
QUESTION [17 upvotes]: In category theory, a subobject of $X$ is defined as an object $Y$ with a monomorphism, from $Y$ to $X$. If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic? It is not true in general that having monomorphisms going both ways between two objects is sufficient for isomorphy, so it would seem the answer is no.
I ask because I'm working through the exercises in Geroch's Mathematical Physics, and one of them asks you to prove that the relation "is a subobject of" is reflexive, transitive and antisymmetric. But it can't be antisymmetric if I'm right...

REPLY [4 votes]: The free group on two letters contains as subgroups groups that are isomorphic free group on any finite number of letters.  The free group on $n \geq 2$ letters contains the free group on two letters as a subgroup.  So if we consider the category of groups, with $A = F_2$ and $B = F_n$ ($n > 2$) we get a counterexample.