The short answer is that a category also satisfies two rules: every object has an arrow to itself (identity) and if there are two arrows (a->b) and (b->c) there is an arrow (a->c) (associativity).
Other abstractions such as functors (mappings between categories) are built upon this relatively simple foundation. You can also think about categories of categories (they can be considered an object, after all!), and so on and so forth.
Some complain about the generality of Category Theory ('It is a theory about everything and nothing!'), but its generality makes it compelling for some for study.
The short answer is that a category also satisfies two rules: every object has an arrow to itself (identity) and if there are two arrows (a->b) and (b->c) there is an arrow (a->c) (associativity).
Other abstractions such as functors (mappings between categories) are built upon this relatively simple foundation. You can also think about categories of categories (they can be considered an object, after all!), and so on and so forth.
Some complain about the generality of Category Theory ('It is a theory about everything and nothing!'), but its generality makes it compelling for some for study.