What are "maps" between proper classes?  When defining a functor (between categories), I am usually told that it assigns to each object of the source category an object of the target category. I do not find this very satisfactory since we are dealing with proper classes here. Judging by the definition, it must be possible to have the concept of a "map" between proper classes. I would like to know what exactly that is and how it is defined.
I have attempted to read some books on set theory in search for an answer, but they all treat classes very briefly and never mention the possibility of having anything like a map between two of them. I would be just as happy if you could point me to a book where this is explained.
 A: Your question sounds like your preferred foundation is set theory, so let me speak in terms of set theory. A map $f : A \to B$ between sets is a functional relation, i.e., a subset $f \subseteq A \times B$ satisfying:


*

*Totality: $\forall x \in A . \exists y \in B . (x,y) \in f$

*Single-valuedness: $\forall x \in A . \forall y, z \in B . ((x,y) \in f \land (x,z) \in f \implies y = z)$.


We usually write $f(x) = y$ instead of $(x,y) \in f$.
The same definition applies to classes. A map $F : C \to B$ between classes $C$ and $D$ is a subclass of $C \times D$ which is total and single-valued.
Exercise (allowed since this is not a research question): the domain and codomain of a function $F : C \to D$ cannot be recovered from the functional relation $F$. (If $C$ and $F$ are empty, how do we recover $D$?) Therefore, the object part of a functor must be a triple $(C,D,F)$ rather than just $F$. But how can we form ordered triples of classes?
A: http://en.wikipedia.org/wiki/Ordered_pair#Morse_definition
Definition:

A relation $R$ is functional if and only if for all ordered pairs $\langle x,z\rangle$ and $\langle y,w\rangle$ in $R$, if $x=y$ then $z=w$.
Definition: If $R$ is a relation, $\operatorname{Range}(R) = \{y : (\exists x)(\langle x,y\rangle \in R)\}$.
Definition: A map is an ordered pair $\langle R,C\rangle$ such that $R$ is a functional relation and $\operatorname{Range}(R) \subseteq C$.
A: Instead of MK set theory + Morse ordered pair definition, is ARC set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, Bibliography PDF) with the usual Kuratowski ordered pair an option? 
ARC supposedly proves the existence of the $n$-th power-class of the set universe V for any $n \in \mathbb{N}$, and all so-called "good" classes provably exist, good classes being the class of all sets and "the powerclass and the union-class of a good class, and the union-class, the intersectionclass, the complement-class, the pair-class, the ordered pair-class, and the Cartesian product-class of any finite number of members of one good class".
According to the cited paper, ARC is consistent relative to ZFC plus a strongly inaccessible cardinal axiom.
I think that this set theory looks quite nice, so I'm wondering why it didn't take off at all.
The formal proofs should be in Muller's PhD thesis, which I don't have access to.
