>what limitations of set theory made it necessary to invent/discover category theory?
Category theory did not start as alternative to set theory.
But: "Category theory even leads to a different theoretical conception of set and, as such, to a possible alternative to the standard set theoretical foundation for mathematics."
>What do categories let us do that we can’t do with sets?
"At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated"
Some category theory constructions like adjoints and monads are higher level and more powerful than basic set theory constructions like power set.
"The number of mathematical constructions that can be described as adjoints is simply stunning."
https://plato.stanford.edu/entries/category-theory/
>what limitations of set theory made it necessary to invent/discover category theory?
Category theory did not start as alternative to set theory.
But: "Category theory even leads to a different theoretical conception of set and, as such, to a possible alternative to the standard set theoretical foundation for mathematics."
>What do categories let us do that we can’t do with sets?
"At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated"
Some category theory constructions like adjoints and monads are higher level and more powerful than basic set theory constructions like power set.
"The number of mathematical constructions that can be described as adjoints is simply stunning."