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Let $C$ be a category internal to a category $K$. It is well known (for example see Proposition 2.4 in the paper Higher Dimensional Algebra VI: Lie 2-Algebra by Baez and Crans https://digitalcommons....

Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...

I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are ...

In the book "Topos Theory" of Peter Johnstone (Topos Theory, LMS Monographs no. 10. Academic, 1977) one finds at page 41 in Chapter 2: "For a detailed account of internal categories ...