The most recommended introductory books on category theory (at least for pure mathematicians) are probably those by Steve Awodey, Tom Leinster, and Emily Riehl. All three have very considerable virtues.
This is a short, animated visual proof demonstrating two formulas for the sum of the first n positive cubes, one in terms of consecutive odd numbers and the other in terms of the square of a sum of consecutive positive integers. We end with a bonus visualization of a direct version of Nicomachus's theorem.