the abstraction of equality

Question

In finitely presented groups, we can define equivalence classes simply by writing equations in the generators : $abc=d$. In this equivalence class we find elements like this $a(aa^{-1})bc$. We can interpret these equations on a finite category, where the generators are just the arrows. Then we are basically saying this chain of arrows is equivalent to this other chain of arrows, creating a commutative diagram. I get the feeling we can create an abstraction of this equivalence by replacing the equality sign with an arrow. Then you can have arbitrary diagrams, not just chains, and you can say there is an arrow between the diagrams. This then sounds like an endofunctor on a category, which is defined by how it maps diagrams to diagrams. Is there a name for this kind of abstraction of the notion of equivalence? What is the general theory of this?