What is the difference between Categories and Relations? For a common basis, I'll state basic definitions of a category and the relation type I'm thinking of. They're here for quick clarity, not precision, so feel free to revise for an answer.
Category:
A collection of objects, a collection of arrows each with a source and target among said objects, identity arrows, and a composition operator satisfying an associative law over arrows.
Reflexive relation on AxA:
A collection of objects, a labelled multi-set of pairs of said objects satisfying the reflexive law, and the standard relation combinator operation (which satisfies the associative law).
Note that labelled multi-set relations are commonly expressed and utilized in discrete math as multi-graphs. I collapse graphs and relations here for unified axiomatic treatment. This is not to treat, e.g., paths over graphs as different from relation composition.
What, exactly, separates these two? They appear essentially identical in definition and meaning to me. Relations immediately appear more general, given that categories only analogize to a certain restricted class of relations.
I note some notational difference, such as how categories name each arrow; I don't see how that would change mathematical power. Similarly with the explicit treatment of composition. While such contrivances can come in handy for certain proofs or explorations, I don't see how it justifies treating the two as separate branches rather than syntactic shims on identical concepts.
[EDITS: fixed associativity statement, extended relations to multi-set representation with graph analogy]

This is an example of that the axioms of a theory doesn't say much of the intentions with the theory. The purpose of relation has never been to study universal properties, for example.
