Algebraic vs. Coalgebraic methods

As I started learning category methods, I noticed that my prior attitude dismissing it as "abstract nonsense" was based upon ignorance. As somebody with physics background I begin to appreciate shifting emphasis from objects onto symmetry, invariance (err morphisms). So in order to understand structure one has to master morphisms. Fine.

Let's compare this with algebraic perspective. There we study objects and operations between them. When we constrain those operations with some axioms we get an algebraic system. A significant part of studying a concrete algebra would be discovering its axiom system. For example, Relation Algebra has been discovered by Pierce, but hasn't been fully axiomatized until Tarski.

Now, how would I study Relation Algebra with methods of category theory? As discovery is significant element of algebraic method and category theory is in a sense dual perspective, I'd guess drawing arrows wouldn't advance me too much, and I would have to discover some important morphisms... Perhaps, a helpful analogy would be how is it done in other categories (preferably, from elementary mathematical fields:-)?