Take a category $C$, and take all endofunctors of $C$, so the set $E= \{ M| M: C \rightarrow C \}$. $E$ forms the objects of a category with morphisms given by all natural transformations $\mu : M \rightarrow N$ for $M,N \in E$. Let $\mathcal{C}$ be the endofunctor category as defined. What are the internal categories in $\mathcal{C}$? 


Further suppose $C$ is a symmetric monoidal dagger category, in this case, what are the internal categories in $\mathcal{C}$?


Edit: There has been some talk about adding structure to $[C,C]$. My original attempt to answer this question was based [on this work](https://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category). So, let us suppose $E$ has a monoidal structure given by functor compostion. What can we say then about $Cat[C,C]$. My last attempts to answer this question focused on functors that were exact. What if we restrict $[C,C]$ to the exact functors?



What are the internal categories in an endofunctor category
(I assume here that by “internal category” you mean the usual sense defined in any category with finite limits.)


If $\newcommand{\C}{\textbf{C}}\C$ has finite limits, then for any $\newcommand{\D}{\textbf{D}}\D$, $[\D,\C]$ will have finite limits, constructed as pointwise limits, and so internal categories in $[\D,\C]$ will just be functors from $\D$ into internal categories in $\C$. So in particular, $\mathrm{Cat}[\C,\C] \simeq [\C,\mathrm{Cat}(\C)]$.


