Well, that wasn't very concrete. How about this: can you give an example of a linear-algebraic fact that is more easily shown by reasoning about these diagrams

Well, that wasn't very concrete. How about this: can you give an example of a linear-algebraic fact that is more easily shown by reasoning about these diagrams than by just using the numbers-moving-along-wires interpretation to immediately turn it into regular equations? For example, the only thing I can think of is maybe (AB)^T = B^T A^T.