What's the name of a morphism the morphism category of the category of categories? Let $Cat$ be the category of categories, then its morphism category consists of functors as objects and morphism between functors as morphisms. If we restrict to the case where the two functors $\mathcal{F},\mathcal{G}:A\to B$, then these morphisms between $\mathcal{F},\mathcal{G}$ are the natural transformations. What is the name of a general morphism between two functors $\mathcal{F}:A\to B,\mathcal{G}:C\to D$?
 A: $\require{AMScd}$
I think you are mistaking between two distinct notions (see also the comments).
The first notion is the arrow category, defined as follow. Let $\mathscr C$ be category. The category $\operatorname{Arr}(\mathscr C)$ (also denoted $\mathscr C^{\mathbf 2}$ or $\mathscr C^\rightarrow$) is the category whose


*

*objects are the arrow $f$ of $\mathscr C$,

*morphisms $(f\colon a \to b) \to (g \colon c \to d)$ are the commutative square
$$ \begin{CD}
a @>f>> b \\
@VVV @VVV \\
c @>>g> d ,
\end{CD}$$

*composition is the concatenation of such squares.


You can of course apply that definition with $\mathscr C = \mathsf{Cat}$.
The second notion is the enrichment of $\mathsf {Cat}$ over itself. That is, the category $\mathsf{Cat}$ has the property that, for any two objects $A$ and $B$, the hom-set $\hom_{\mathsf{Cat}}(A,B)$ actually carries a category structure in such a way that the composition
$$ \hom_{\mathsf{Cat}}(B,C) \times \hom_{\mathsf{Cat}}(A,B) \to \hom_{\mathsf{Cat}}(A,C) $$
is a functor. The short way to say it is : $\mathsf{Cat}$ is enriched over the (cartesian closed) monoidal category $(\mathsf{Cat},\times,\mathbf 1)$ (where $\mathbf 1$ is the final category).
The two notions are very distinct and not to be confused !
A: There is a more general notion that might be relevant here: the double category $\Bbb{CAT}$ of categories, functors and profunctors, where a profunctor $\mathcal A\not\to\mathcal B$ can be defined as a category $\mathcal U$ equipped with a functor $U:\mathcal U\to 2$ where $2=\,[x\to y]$ is the category of one arrow and $\mathcal A=H^{-1}(x)$ and $\mathcal B=H^{-1}(y)$ [see e.g. in nLab].
Let the horizontal arrows of $\Bbb{CAT}$ be the profunctors, the vertical arrows the functors, and let the cells -what you are looking for- be the functors $T:\mathcal U\to\mathcal V$ for which $T|_\mathcal A=F$ and $T|_\mathcal C=G$:
$$ \mathcal A \overset{\mathcal U}{\not\to}\mathcal C \\ \ 
F\downarrow\ T\ \downarrow G \\
\mathcal B \underset{\mathcal V}{\not\to}\mathcal D
$$
You can verify that if $\mathcal U$ and $\mathcal V$ are both the identity profunctors, then we get back exactly the natural transformations $F\leadsto G$.
