Concrete examples of internal categories (other than small categories)

Question

I understand the definition of an internal category, but so far I haven't found examples of internal categories (other than categories internal to $\mathrm{Set}$, which are just the small categories) that one could encounter "naturally" in mathematics. I have only found other abstract concepts related to this one. What are some concrete examples that can motivate the study of internal categories? In particular, why is it useful to think of operations such as "assigning the domain to a morphism" as being a morphism in a certain category?

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The only example I've found here is in this answer, namely that where we consider a topological space $X$ as the set of objects of a category, and $X \times X$ with the product topology as the set of morphisms, with the "obvious" operations. This is interesting but seems a bit pointless at first sight; for instance, what is the use of thinking of composition as a continuous operation in this example?

Answer 1

This is no means a comprehensive answer, but I can't post a comment because I don't have enough points.

One example that has been very useful for me is that strict monoidal categories are categories internal to monoids.

This was used by Lack in order to compose props via distributive law.

Also, double categories are categories internal to Cat.

Answer 2

To every group object $G$ (internal group) in a category with finite limits, we can associate an internal category: let $1$ be the object of objects and $G$ be the object of morphisms, and composition is multiplication in $G$.