You can present a group in a Cayley-like manner, replacing colors by explicit assignment of nodes to edges: while in a Cayley graph $x \circ y = z$ is presented like this:

you can also present it like this:

Now the group axioms can be stated like this:

• For each $x, y$ there is a unique $z$ with $x \circ y = z$, or: such that the following diagram holds ("commutes"):

• There is an $e$ such that for all $x$ it holds that $x\circ e = e \circ x = x$, or: such that the following two diagrams commute:

and for each $x$ there is a $x^{-1}$ such that $x \circ x^{-1} = x^{-1} \circ x = e$, or: such that the following diagram commutes:

• For each $x, y, z$ it holds that $x \circ (y \circ z) = (x \circ y) \circ z$, or: such that the following diagram commutes:

The last diagram is somewhat ugly, even when drawn in this most balanced way (I didn't find a more appealing and symmetric one).

But an astonishing symmetry arises, when we consider Abelian groups. Commutativity is expressed by the diagram:

and associativity becomes:

In the presence of commutativity, associativity seems to be related to commutativity (some sort of "second level commutativity").

Can any use be made of this kind of diagrams, or is it just vain baublery?