"Functors" which map objects to morphisms

Question

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!

Answer 1

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.

Answer 2

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.