Why this episode OP? IMO the proof of fullness is quite belabored. In fact the very next episode gives a much better one-page proof in the section starting "Her

Why this episode OP? IMO the proof of fullness is quite belabored. In fact the very next episode gives a much better one-page proof in the section starting "Here’s what we mean by this rather cryptic remark". It's faithfulness that's more interesting, ie. are the graphical rules sufficient to prove that any two diagrams that produce the same outputs given the same inputs are equal, ie. is the list of graphical rules complete? A string diagram is also equivalent to a program in a generalized-stack language (in the stronger sense of equivalence up to difference in the topological arrangement of the wires, not up to the full graphical rules). Unlike a pure stack language, you can operate on values at any point in the stack, not just the top. The four operations are: drop an element from the stack, push a zero to the stack, remove two elements from the stack and push their sum to the stack, remove an element from the stack and push two copies of it to the stack. Obviously any program in this language computes a linear function of its initial stack (there is a full homomorphism to matrices). A stack language captures the same ability of an input to be consumed that a string diagram does.