We give groups their axioms of closure, associativity, inverse, and identity, so we can "do algebra" or equivalently, study symmetries in a very general sense. What new can we do by adding the requirement that a category contain arrows from an object to itself? Surely this removes some level of generality, which I think the idea of creating categories was seeking after?
I know there are many examples of categories, and that this is used to show the importance of them. But I'm sure there's a reason we use categories in each case, and that there must be at least a vague thread that goes through all of them. I'm not so interested in specific applications, if possible but if so that's fine.
QUESTION: by giving category objects unique identity arrows what useful (but AS GENERAL AS POSSIBLE) property is gained? Since I'm not familiar with this part of math I'd appreciate something that's intuive and in layman's terms. :) Thanks.