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?

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    $\begingroup$ Do you want $C$ to have finite limits (so it is cartesian together with $E$, and composition of internal categories as a map from the pullback of source versus target), or instead $E$ to be monoidal wrt composition (I bet internal monoids here are pretty strange animals)? $\endgroup$ – Fosco Loregian Jul 15 '18 at 7:36
  • $\begingroup$ @FoscoLoregian: Internal monoids in $[C,C]$ w.r.t. composition are just monads on $C$ — very familiar and friendly animals! But yes, as you say, the poster needs to be clear about what sense of “internal categories” they mean: just the general one in any category with finite limits (in which case the answer is a bit boring)? or one of the more elaborate senses, defined in a category with extra structure, in which case they need to explicate what extra structure on $[C,C]$ they have in mind? $\endgroup$ – Peter LeFanu Lumsdaine Jul 15 '18 at 8:31
  • $\begingroup$ Another thing the OP might have in mind, I guess, is categories enriched in $[C,C]$ (with the composition monoidal structure). These would then be a fairly straightforward generalisation of monads — but it’d be interesting to see examples! $\endgroup$ – Peter LeFanu Lumsdaine Jul 15 '18 at 8:42
  • $\begingroup$ Of course I wanted to write "internal categories", but somehow overwritten it to "monoids" :-) $\endgroup$ – Fosco Loregian Jul 15 '18 at 8:53
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    $\begingroup$ @FoscoLoregian: Ah, OK. But there again, there’s no standard definition of “internal categories in a monoidal category” (as far as I’m aware): you need to put some more elaborate structure on the ambient category, since internal categories need pullbacks not just products. $\endgroup$ – Peter LeFanu Lumsdaine Jul 15 '18 at 13:49

(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, $\mathrm{Cat}[\C,\C] \simeq [\C,\mathrm{Cat}(\C)]$.


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