Arrow of Arrow Set is a Functor? My question is whether arrows between arrows (or arrows in $\bf{Set}^{\rightarrow}$) are, by definition, functors who, in the general sense, map from $\bf{Set}$ to itself.
The definition of a functor is...
For categories C, D and a functor $F: \bf{C}\rightarrow \bf{D}$, then $F$:


*

*associates each object $x\in \bf{C}$ an object $F(x)\in \bf{D}$

*associates each arrow $f: x\rightarrow y$ in C an object $F(f): F(x)\rightarrow F(y)$ in D such that identities and composition are preserved.


An arrow in $\bf{Set}^{\rightarrow}$ takes one function $f: x\rightarrow y$ to another function $g: w\rightarrow z$, which are both in $\bf{Set}$. Since identities and composition must be preserved within a category, then any arrow in $\bf{Set}^{\rightarrow}$ will also preserve identities and composition both in terms of $\bf{Set}^{\rightarrow}$ and in terms of $\bf{Set}$ by acting on the arrows (that also preserve identities and composition) in $\bf{Set}$.
I can't see a contradiction here. Any thoughts? 
Thanks in advance.
 A: An arrow between arrows $f,g$ is just a commutative square, so two arrows $h,k$ such that $hf=gk$. In particular, only four sets are involved, whereas a functor on the category of sets must choose an image for every single set. There's no relationship, unless I misread you.
A: A category needs both objects and arrows. In fact, it is quite common to actually forget about the objects of a category, so we work simply need arrows. All I want to say is that the arrows of a category should be defined before anything else.
Given a category $\mathbf{C}$, we can define its category of arrows, which you denote $\mathbf{C}^\rightarrow$, but is more commonly denoted $\mathbf{C}^2$. For this, we need to define the arrows in $\mathbf{C}^2$ (and the objects, if possible). It does not make sense to ask what are the arrows in $\mathbf{C}^2$, since they should be explicitly defined beforehand.
So, by definition: the objects of $\mathbf{C}^2$ are arrows of $\mathbf{C}$; Given two objects $f,f'$ in $\mathbf{C}^2$, i.e., two arrows $f:a\to b$, $f':a'\to b'$ in $\mathbf{C}$, an arrow $f\to f'$, in $\mathbf{C}^2$, is a pair $(g,h)$ of arrows in $\mathbf{C}$, with $g:a\to a'$ and $h:b\to b'$, satisfying $hf=f'g$. This is the same as the followin diagram commuting
\begin{array}{rcc}
a&\overset{\small f}{\longrightarrow}&b\\
{\small{g}}\left\downarrow\vphantom{\int}\right.&&\left\downarrow\vphantom{\int}\right.{\small{h}}\\
a'&\underset{\small{f'}}{\longrightarrow}&b'
\end{array}
Arrows, and objects, do not need to satisfy any property (other than having a composition, etc...). We usually think of functors as morphisms between categories (i.e., they preserve their structure/composition), so functors work as arrows in the category $\mathbf{Cat}$ of all categories. This is very different from arrows between arrows.
