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. 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?

enrichedin $[C,C]$ (with the composition monoidal structure). These would then be a fairly straightforward generalisation of monads — but it’d be interesting to see examples! $\endgroup$2more comments