In category theory, a subobject of object $A$ is defined to be an equivalence class of isomorphic monomorphisms into $A$. Does this seem weird to anyone else? Isn't it normal to allow something to be only defined "up to isomorphism"? Sure, we could define a product to be the equivalence class of objects satisfying the universal property, but then it wouldn't live in our category. And it may well be a proper class. No one defines limits this way, why do we do this for subobjects and quotient objects?
If we just defined a subobject of $A$ to be a monomorphism into $A$, then the class of subobjects of $A$ would only be a preorder, instead of a poset. So what?
Best Answer
The question is ancient, but IMHO the most convincing answer is missing:
The definition of "subobject in a category $\mathcal{C}$" is chosen in such a way that it generalize the notion of $k$-vector subspaces (when $\mathcal{C} = \mathsf{Vect}_k$), the notion of subsets (when $\mathcal{C} = \mathsf{Set}$), the notion of subrings (when $\mathcal{C} = \mathsf{Ring}$), and lots of other classical "sub-something" notions that appear throughout mathematics (probably not all of them, though). If you were to define "subobjects of $A \in \mathcal{C}$" to mean "monomorphisms with codomain $A$", then the notion of a subobject would not generalize all of these classical notions, because you get too many different subobjects that correspond to the same "sub-something". For instance, in $\mathsf{Set}$, the monomorphisms $\emptyset \to \left\{1\right\}$, $\left\{1\right\} \to \left\{1\right\}$ and $\left\{2\right\} \to \left\{1\right\}$ would be three different subobjects of the object $\left\{1\right\}$, but there are only two subsets of the set $\left\{1\right\}$. So this would be a bad definition.
However, if you define subobjects of $A \in \mathcal{C}$ to be isomorphism classes of monomorphisms with codomain $A$ (where "isomorphism" is to be correctly interpreted: an isomorphism between two monomorphisms $\alpha : S \to A$ and $\alpha^{\prime} : S^{\prime} \to A$ means a morphism $s : S \to S^{\prime}$ satisfying $\alpha = \alpha^{\prime} \circ s$), then, in all of the examples listed above, the subobjects of $A$ are in a canonical bijection with the "sub-somethings" (i.e., the $k$-vector subspaces, or the subsets, or the subrings). For instance, in $\mathsf{Set}$, the subobjects of a set $A$ are the isomorphism classes of monomorphisms with codomain $f$. The isomorphism class of such a monomorphism $f : S \to A$ can be identified with the subset $f\left(S\right)$ of $A$. Thus, the subobjects of $A$ are in bijection with the subsets of $A$ here. The same construction works for rings and for $k$-vector spaces.
The good definition of a subobject also has the advantage (compared with the bad definition) that the subobjects of a given object $A \in \mathcal{C}$ often form a set (as opposed to just a class). I don't personally find this vital; it is not usually true in constructive mathematics anyway, and I don't believe that a definition is necessarily bad just because it sometimes returns proper classes.