As you may have guessed from the title I'm talking about category theory (CT). It's a fascinating subject to me. It can beautifully describe the essence of what it means to "have a structure" and it's a wonderful tool for studying mathematics itself.
The definition of a category is really really simple (objects, morphisms, and some rules of composition and associativity). Nonetheless you can have very sophisticated theorems when using multiple nested layers (so called n-Category).
So my question is: Since it seems arbitrary to use composition of morphisms between objects to describe what it means to have some kind of structure, have we come to this logically? Or was it an arbitrary choice (that fortunately turned out to work very well)? Are there any attempts to go deeper in abstraction or is there a good reason to think that CT is really fundamental? Is there a way of building complexity from simplicity with something other than morphisms and compositions, and which would be at least as useful as CT?