I believe they are higher dimensional string diagrams.
Maybe called n-diagrams? They are used in higher homotopy and higher category theory, I believe.
But not entirely sure.
Note the conclusion at the bottom, the proof on the right and the axiom on the left doesn’t seem to be related.
The proof on the right is Theorem 6; the equality at bottom is in section 3.4, where the proof is omitted because “follows from definition”; the axiom on the left is HM1 and HM2 on page 19.
What are we seeing here?
I believe they are higher dimensional string diagrams. Maybe called n-diagrams? They are used in higher homotopy and higher category theory, I believe. But not entirely sure.
https://arxiv.org/pdf/2305.06938
EDIT: Found it! they are called surface diagram, which are generalization of string diagram to 3-categories https://golem.ph.utexas.edu/category/2010/03/modeling_surface_diagrams.html https://ncatlab.org/nlab/show/surface+diagram
Still not sure what the proof is talking about though :(
But from the conclusion it looks like some sort of natruality condition, where the morphisms are slided around except beta.
EDIT AGAIN: got in touch with my string diagram contact. Here is the paper https://arxiv.org/pdf/0807.0658
Note the conclusion at the bottom, the proof on the right and the axiom on the left doesn’t seem to be related.
The proof on the right is Theorem 6; the equality at bottom is in section 3.4, where the proof is omitted because “follows from definition”; the axiom on the left is HM1 and HM2 on page 19.