The special role of certain Sets in the theory of sets I was thinking about the two element set today, and how it can be used to represent a binary function.  One very important binary functions is whether or not an element in set $A$ is in another set $B$.  A map from $A$ to $2$ could define membership in set $B$.  So, the two element set $2$, is important for membership.  How are other sets used in the semantics of set theory itself?  What is the abstract theory of this phenomenon?
A commenter has pointed out that I am talking about subobject classifiers.  These are based on the limit of the cospan diagram $X \rightarrow 2$ and $1 \rightarrow 2$.  Is there a heirarchy of "internal concepts" that we can construct with sets of increasing cardinality using composition of cospans and their limits?
 A: The abstract theory of this phenomenon deals with representable functors.
Let $\mathbf{Set}$ denote the category of sets. Given a category $C$ and a functor $F\colon C^{\mathrm op}\to \mathbf{Set}$ we can ask if it is representable, i.e. whether it is naturally isomorphic to a functor $H_a=\operatorname{Hom}(-,a)$ for an object $a$ in $C$.
In your example, $F$ is the functor $\mathbf{Set}\to\mathbf{Set}$ which sends a set to its powerset. This functor is representable by the $2$-element set.
In general, we think of a functor $F\colon C^{\mathrm op}\to \mathbf{Set}$ as assigning to every object $c\in C$ a set of "structures $S$ of a certain  type on $c$". Being representable by an object $a\in C$ then means that there is a "universal structure $S_{\mathrm{univ}}$ of that type on $a$" in the sense that every other structure $S$ on $c$ arises as the pullback of $S_{\mathrm{univ}}$ along a unique morphism $c\to a$.
Let me give a more non-trivial example: Fix sets $x,y$ and consider the functor $\mathrm{Set}\to\mathrm{Set}$ given by $c\mapsto \operatorname{Hom}(c\times x,y)$. This functor represented by the set $y^x$ equipped with the universal element of $\operatorname{Hom}(y^x\times x, y)$ given by the evaluation map $y^x\times x\to y$.
Of course one can also dualize and consider functors $C\to \mathrm{Set}$ which are corepresentable (i.e. isomorphic to $\operatorname{Hom}(a,-)$). A very basic example would be the identity $\mathrm{Set}\to\mathrm{Set}$ which is corepresented by any singleton set.
