TITLE: Categorial definition of subsets
QUESTION [10 upvotes]: I cannot see why this definition http://en.wikipedia.org/wiki/Subobject 
is equivalent to subsets in the category of sets. I am confused by the following facts, 
1) the partial order is defined between morphism, and not objects (which are the sets in the category of sets), but the subset relation must hold between sets (i.e. objects)
2) why are there three objects involved, when for a subsetrelation just two sets are involved?
How does this defintion makes sense for example for $\{ 1 \} \subset \{ 1,2,3 \}$?

REPLY [2 votes]: 1) the partial order is defined between morphisms, and not objects
  (which are the sets in the category of sets), but the subset relation
  must hold between sets (i.e. objects)

There is an unfortunate notational confusion here, which experienced category theorists untangle mentally, but it helps to spell out explicitly for beginners.
When we write $u : S \to A$, we mean that there is a morphism $\langle S, A, u \rangle$ in the collection $\textbf{Mor}_C$ of the category.  In other words, the morphisms carry with them their own source and target objects (the $S$ and the $A$ in this case).  So, as you can see, when we are defining a partial order on morphisms, the source objects $S$ are also involved.  (Using triples $\langle S, A, u\rangle$ is one way of formalizing what morphisms are.  There are also other ways, but they will get more complicated in this context.)
Since the partial order is being defined only between morphisms that lead into $A$, we can mentally erase the $A$ in the data, and regard the morphisms as just being pairs $\langle S, u\rangle$.  So, we are saying $S$ is an object that "looks like" a subset of $A$, and $u$ is the way to interpret its elements as the elements of $A$.
So, $\langle S, u\rangle \leq \langle T, v\rangle$ means that $S$ and $T$ "look like" subsets of $A$, we have ways of interpreting their elements as those of $A$ and, further, there is also a monomorphism $w : S \to T$ which allows us to interpret the elements of $S$ as the elements of $T$ in a way that is consistent with $u$ and $v$.
Moreover, $\langle S, u\rangle \equiv \langle T, v\rangle$ means that $S$ and $T$ are isomorphic in a way that is consistent with the chosen interpretations $u$ and $v$.

2) why are there three objects involved, when for a subsetrelation
  just two sets are involved?

You want to think of the objects in a category as types rather than as sets.  You can only talk about the equality of two elements within the same type.  So, the normal set-theoretic locution $S \subseteq T \iff (\forall x.\, x \in S \to x \in T)$ is incoherent from a types point of view.  It involves taking an element of type $S$ (viz., $x$) and talking about its equality with the elements of type $T$.  A type-correct way of saying $S \subseteq T$ is 
$$\forall x : S.\, \exists y : T.\, u(x) =_A v(y)$$
In other words, to talk about $S$ being a subset of $T$, we need a larger set $A$, of which both $S$ and $T$ are part (intuitively), and we have ways of interpreting the elements of $S$ and $T$ as elements of $A$ (provided by the monomprhisms $u$ and $v$.)  This is precisely what is meant by $\langle S, u\rangle \leq \langle T, v\rangle$.