I am sad to bring sad tidings but... This is very beautiful, it just puts the burden of the proof on a more abstract setting, which may be more difficult to grasp.
(This is just a comment on the "graphical proofs", they are beautiful but they cannot be "proofs" unless you provide the background).
These are not "graphical proofs" in the sense of pictures that suggest patterns in mathatical relations. This is (or can be) a formal graphical language for (a branch of) mathematics. Like any formal language it's not self evident how or why it works and needs to be rigourously defined.
(This is just a comment on the "graphical proofs", they are beautiful but they cannot be "proofs" unless you provide the background).