*writing* proofs involving commutative diagrams

"As I said above, I know (some way) to show this, it involves a mess of commutative diagrams, and writing this out would take a remarkable amount of space for a sequence of trivialities."

Maybe that's okay to begin with. As you become more adept at the subject you can start leaving out details, but writing a sequence of commutative diagrams, while time and space consuming, is not that different from showing two knots are the same by writing down an explicit sequence of Reidemeister moves. In both cases, it seems like there is no simpler way of doing it.