Algebraic manipulations in monoidal categories can also be performed in a graphical calculus. And the best part is that this is completely rigorous: a statement holds in the graphical language if and only if it holds (in the algebraic formulation). See for example Peter Selinger's "[A survey of graphical languages for monoidal categories][1]". There are many instances, for example in knot theory studied via braided categories. The following specific example comes from Joachim Kock's book "[Frobenius Algebras and 2D Topological Quantum Field Theories"][2], and proves that the comultiplication of a Frobenius algebra is cocommutative if and only if the multiplication is commutative.

![alt text][3]


  [1]: http://dx.doi.org/10.1007/978-3-642-12821-9_4
  [2]: http://books.google.com/books?id=6dZZW08Z04MC
  [3]: http://oi55.tinypic.com/5k58uf.jpg