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?



The Abstraction of Equality
I'm not quite sure what you mean, but one possibility for what you may be driving at is the following. It may seem like an "arrow of diagrams" should be a functor, but usually the categorification of "imposing an equality" is not acting by a functor but rather "adding a morphism". We can thus categorify a group presentation in the following steps:


1. Present a category rather than a group. Now the generators are arrows rather than elements, and the words are strings of composable arrows, and the relations are, as you said, equalities between parallel such strings, i.e. diagrams that are stipulated to commute.
2. Present a 2-category rather than a category. Now the generators become the 1-morphism generators, and the relations are replaced by "2-morphism generators": instead of stipulating that some diagram commutes, we freely insert a 2-morphism whose domain is one side of the diagram and whose codomain is the other side. Then we can assert a third level of "relations" between the pasting composites of these 2-morphisms.
3. Present a 3-category rather than a 2-category. Now the third level of relations is replaced by "3-morphism generators", which can be thought of as "arrows between pasting diagrams".
4. Ad infinitum.


The general theory of this sort of thing goes by the name of [computads](https://ncatlab.org/nlab/show/computad).


