Often discovering the right way to write something makes it hugely easier to think about it. Examples include many musical notations, the Leibnitz notation for calculus, and Feynman diagrams in physics. A 2007 paper "A Not-So-Characteristic Equation: The Art of Linear Algebra" by Elisha Peterson leads to a new (old?!) rabbit-hole visual-language of Penrose graphical notation that promises potential enlightenment in other areas, including bits of category theory. Peterson suggests:

The emphasis of this paper is on illumination rather than proof, and in particular on how diagrammatic techniques have the power to both prove and explain. For this reason, several examples are included, and more enlightening proofs are preferred. While diagrammatic methods may seem unfamiliar at first, in the end they offer a profound insight ...

We hope that by the end of the paper the reader is both more comfortable with the appearance of diagrams in mathematics and convinced that sometimes, in the words of Richard Feynman, "these doodles turn out to be useful."

(cf BraKet (2001-01-24), Fractal Feynman (2003-01-30), Greatest Inventions (2011-06-09), Category Theory Concepts (2016-04-25), Why Care about Category Theory (2019-03-03), Structure Itself (2019-03-22), Applied Category Theory (2019-04-24), ...) - ^z - 2019-12-10