When we study the categorical theory, to check the commutativity is a routine work.
Using a string diagrammatic notation, the commutativity is replaced by more intuitive gadgets, the elevator rules.
I choose the Yoneda lemma as a mile stone of categorical theory, and will explain the equation-based proof using the string diagrams.
1: Category theory: a programming language-oriented introduction (Pierre-Louis Curien)
(especially in section 2.6)
You can get the pdf file in the below link:
2: The Joy of String Diagrams (Pierre-Louis Curien)
3: (in progress) Cat (Ray D. Sameshima)
4: Physics, Topology, Logic and Computation: A Rosetta Stone (John C. Baez, Mike Stay)
If you are physicist, this is a good introduction to category theory and its application on physics.
His string diagrams, however, differ from our one little.
5: Category Theory Using String Diagrams (Dan Marsden)
1 Category, functor, and natural transformation
3 String diagrams
4 Yoneda lemma and string diagrams