"Functors" which map objects to morphisms I'm just beginning to learn category theory, and this question popped up in my mind, which Googling has not been able to resolve (probably because I am not searching the right terms).
A functor between categories $C,C'$ is a map $T$ that associates each object $c\in C$ with an object $Tc\in C'$, and each morphism $\varphi$ in $C$ with a morphism $T\varphi$ in $C'$, satisfying certain niceness properties (takes identity morphism to the identity, etc.)

What about a map $T$ from $C$ to $C'$ that takes objects in $C$ to morphisms in $C'$, and morphisms in $C$ to "morphisms between morphisms" in $C'$ instead, satisfying similar "niceness" properties? Do these have a name?

This is in some sense like a "meta" version of functors, and I imagine they have been studied before. I imagine that higher category theory might be relevant, but I unfortunately do not quite understand the Wikipedia page well enough to determine whether or not it's related.
Unless, perhaps, these maps are not well-defined, or not interesting? If so, I am very intrigued to find out why. Indeed my question above is rather vague as I do not really know how to make the notion of transformations between morphisms rigorous. (Maybe it is captured by the notion of a natural transformations? Or, maybe not.)
Any explanations or directions on where to find out more, or the right terms to search, would be greatly appreciated!
 A: In some cases you can map object to a functor such as the famous Yoneda lemma in CT.

If $\mathcal {C}$  is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object $A$ of $\mathcal {C}$  gives rise to a natural functor to $\mathbf {Set}$ called a hom-functor.


The original category $\mathcal {C}$  is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in $\mathcal {C}$... The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner.

But it's impossible to have a well formed map $T$ from $C$ to $C'$ that takes objects in $C$ to morphisms in $C'$ in general. Since usually there may be multiple or infinite morphisms in $C'$ related with any target object in $C'$ from a source object in $C$, then how you can define such a map from a source object to so many target morphisms? You miss the necessary link between your conceived source and target. In contrast, Yoneda lemma mapping a source object to a functor preserves this link.
A: For any category $D$ there is its category of arrows $Arr(D)$. Its objects are the morphisms in $D$, and its morphisms are commuting squares of morphisms in $D$. A functor $C\to Arr(D)$ thus takes the objects of $C$ to morphisms in $D$, and morphisms in $C$ to morphisms between morphisms (at least in a sense). This does not require any higher dimensional considerations.
