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The category SET has an internal category, which is a small category with small objects and small morphisms, and that means that it's a subcollection of the collection of objects in SET.
Is an internal category the same as a subcategory? Are they different only when dealing in the context of Grothendieck universes?

Thanks!

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No. An internal category in $\text{Set}$ is just a (small) category. A subcategory of $\text{Set}$ is a category equipped with an inclusion functor into $\text{Set}$.

The difference between the two constructions is that in the first case you pick out two objects in $\text{Set}$, one to serve as objects and one to serve as morphisms in your internal category. In the second case you pick out a whole bunch of objects, all of which are objects in your subcategory, with morphisms given by some morphisms between them in $\text{Set}$.

It will be easier to make this distinction by replacing $\text{Set}$ with a more general category $C$ (with enough pullbacks). For example, an internal category in $\text{Top}$ is a category whose object and morphism sets are both equipped with a topology such that everything in sight is continuous, whereas a subcategory of $\text{Top}$ is a collection of topological spaces and a collection of morphisms between them containing identities and closed under composition. In particular, it is an ordinary category equipped with an inclusion functor into $\text{Top}$.

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A subcategory of a category $\mathcal C$ is a priori not an internal category of $\mathcal C$. But it can be an internal category of another category $\mathcal C'$.

More specifically for your question about sets : let $U$ be a Grothendieck universe ; then the category $U\!-\!\mathrm{Sets}$ of $U$-small sets is a subcategory of itself (obviously) while it can be an internal category of $U\!-\!\mathrm{Sets}$ (as the set of $U$-small sets is not $U$-small).

However, there exists a Grothendieck universe $U'$ such that you can realise $U\!-\!\mathrm{Sets}$ as an internal category of $U'\!-\!\mathrm{Sets}$.

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