Trying to understand how Lawvere's [E]lementary [T]heory of the [C]ategory of [S]ets can be used as a foundation for mathematics alternative to ZFC, I am getting stuck with the question on how to develop category theory within ETCS.
It's neither a problem in ZFC nor in ETCS to define the notion of a category, at least internally, when objects and morphisms are "sets". However, what is problematic in developing category theory completely internal to ZFC, i.e. without use of proper classes and metatheorems, is the choice of what we want to call the category of sets inside ZFC. One way to solve this is to introduce Grothendieck universes and to take them as the "base-categories of sets", on which category theory within ZFC can be built.
My question now is: What are the possibilites in ETCS to ensure a reasonable "category of sets" based on which category theory can be developed inside ETCS?
Are there any good sources for this question?