Reading through Jonathan's answers, it seems like what natural transformations are doing is overlaying each instantiation of graphs to find similarities and if there is any leftovers in the domain then it folds it so that edges line up in the most natural way kind of like origami where the rule is you have to match corners ie no half folds meaning you must match points to points and edges to edges and not points to edges.

So there are two ways you can "fold" : 1. Matching edge to edge which is more like folding 2. Matching a sequence of arrows to identities which is more like pinching and twisting off. Hope this is making sense LOL...

Not sure but I think when you fold you can't fold the arrows countercurrent. If this happens, you must pinch?