In enriched category theory over a base monoidal category $(\mathcal{V},\otimes\_{\mathcal{V}},\mathbf{1}\_{\mathcal{V}})$, one can consider $\mathcal{V}$ itself as a $\mathcal{V}$-enriched category when it has a closed monoidal structure $[-,-]\_{\mathcal{V}}$.


Is there a similar procedure in internal category theory? That is, starting from a category $(\mathcal{E},\times\_{\mathcal{E}},\mathbf{1}\_{\mathcal{E}})$ with pullbacks and a terminal object, can one associate an $\mathcal{E}$-internal category to $\mathcal{E}$ itself?



Internalising the base in internal category theory
Internal categories are too limited for self-internalisation.
Think of the most fundamental category, $\textbf{Set}$: an internal category is a small category, and $\textbf{Set}$ is (usually) not even essentially small.
(In NF and related set theories with a universal set, $\textbf{Set}$ has a set of objects but fails to be cartesian closed, so I think even then it cannot be self-internalised.)


However, there is a notion of locally internal category – a special kind of fibred (= indexed) category – and the codomain fibration (= self-indexing) of a category with finite limits is locally internal if and only if it is locally cartesian closed.


One thing that bugs me is that there is one feature of locally small categories that does not seem to be captured by locally internally categories, namely that any locally small category is a union of small categories.
But I am not sure whether there is any merit in trying to formalise this in the language of fibred categories.


