A sub-object of an object $A$ in a category is an equivalence class of monomorphisms $s:U \to A$, where $s$ is equivalent to another monomorphism $t:V \to A$ if there exist morphisms $x:U \to V$, $y:V \to U$ such that $$ s = t \circ x, ~~~~~~~~~\text{ and } t = s \circ y. $$
Consider now the alternative definition, where we require $x$, or equivalently $y$ to be isomorphisms. This is clearly more restrictive. What is a good example to illustrate the fact the the first definition is the more natural one to consider?
Actually, the two definitions are equivalent.
Given the existence of those morphisms $x$ and $y$, substituting $t= s \circ y$ into the equation $s = t \circ x$ gives $s= s \circ y \circ x$. Since $s$ is a monomorphism, we have that $y \circ x = id_U$.
Similarly, we can prove $x \circ y= id_V$.
Therefore $x$ and $y$ are isomorphisms with each other as their inverse.