How to write diagram chases
Diagram chasing arguments are supposed to be impossible to write in such a way that it takes somebody less effort to read than to reproduce the argument for themselves. Many years ago, I cooked up a solution to this problem. In fact, this problem was what motivated me to learn more about the Salamander lemma.
Draw the diagram of sets which you're going to chase. The elements being chased will be drawn in a "parallel" diagram, where their relative positioning indicates which set they belong to (see the examples below). Then draw elements connected by arrows using the following rules:
0. The "starting element" should be visually distinguished in some way (I like to put a circle around it)