Action of functors between functor categories I cannot understand if there is a way to understand the action of a functor between two functor categories, "abstractly".
I mean this: let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{C}$', $\mathcal{D}$' be categories and consider the functor categories $Fnct(\mathcal{C},\mathcal{D})$ and $Fnct(\mathcal{C}',\mathcal{D}')$. Let $H : Fnct(\mathcal{C},\mathcal{D}) \longrightarrow Fnct(\mathcal{C}',\mathcal{D}')$ be a functor. How can I understand the action of the functor $H$ on objects and morphisms without any information about what categories $\mathcal{C}$, $\mathcal{D}$, $\mathcal{C}$', $\mathcal{D}$' actually are?
I found this question Functors Between Functor Categories, but it does not exactly answer to mine.
Thank you
 A: There is an important special case.
Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{C}'$, $\mathcal{D'}$ be categories, $T\colon\mathcal{C}'\to\mathcal{C}$ and $S\colon\mathcal{D}\to\mathcal{D'}$ be functors. Then we can define the functor $S^T\colon\mathcal{D}^{\mathcal{C}}\to\mathcal{D}'^{\mathcal{C}'}$ in the following way: $S^T(R)=S\circ T\circ R$ and $S^T(\alpha)(c)=S(\alpha(T(c)))$ for every functor $R\colon \mathcal{C}\to\mathcal{D}$, natural transformation $\alpha\in Arr(\mathcal{D}^{\mathcal{C}})$ and every object $c\in\mathcal{C}$. You can define this functor for every four categories and two functors between them.
For example, let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be categories, $T\colon\mathcal{A}\to\mathcal{B}$ be a functor. Then $T^{I_C}\colon\mathcal{A}^{\mathcal{C}}\to\mathcal{B}^{\mathcal{C}}$ is called a direct image functor and is denoted by $T_*$. Also, the functor
$I_C^T\colon\mathcal{C}^{\mathcal{B}}\to\mathcal{C}^{\mathcal{A}}$ is called an inverse image functor and is denoted by $T^*$.
But, it should be pointed out, that it is of course not true, that every functor $F\colon\mathcal{D}^{\mathcal{C}}\to\mathcal{D}'^{\mathcal{C}'}$ may be presented as $F=S^T$ for some $T\colon\mathcal{C}'\to\mathcal{C}$ and $S\colon\mathcal{D}\to\mathcal{D'}$ (it is not true even for sets), so the functor $(-)^{(-)}\colon\mathbf{Cat}^{op}\times\mathbf{Cat}\to\mathbf{Cat}$ is not full.
A one more special case. Let the first power is trivial, i.e. we consider functors of the form $\mathcal{A}\to\mathcal{C}^{\mathcal{B}}$. Then we may equivalently consider them as bifunctors $\mathcal{A}\times\mathcal{B}\to\mathcal{C}$, because of the exponential isomorphism $(\mathcal{C}^{\mathcal{B}})^{\mathcal{A}}\to\mathcal{C}^{\mathcal{A}\times\mathcal{B}}$.
So, as we can see, there is no (or I don't know) good ways to represent all functors between functor categories, but there are many interesting special cases when such ways exist.
Edit. After some thought, I decided to include the following presentation of all functors between functor categories (in spirit of the second special case): functor $\mathcal{D}^{\mathcal{C}}\to\mathcal{D}'^{\mathcal{C}'}$ is a functor $\mathcal{C}'\times \mathcal{D}^{\mathcal{C}}\to\mathcal{D}'$. But it's hard to call it a simplification.
