TITLE: What's the name of a morphism the morphism category of the category of categories?
QUESTION [5 upvotes]: 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 $\mathcal{F}:A\to B,\mathcal{G}:C\to D$?

REPLY [5 votes]: 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 $F\leadsto G$.