Of course, the below is irrelevant to understanding the proof. However, I am completely mystified by the page 1 sentences: "We now show that f is L-canonical. We shall apply the definition of f repeatedly without referring to this fact." It's only the second of those sentences that confuses me. The first sentence is given for context.
What is the "fact" in question. If the fact is that f is being applied, then of course when you write fP, you're applying f. Why on earth would this need to be pointed out? If the fact is that f is L-canonical, that also makes no sense at all. The statement of f being L-canonical is one half of an A implies B statement. So, for half of the result, we clearly assume it. And for the other half, we deduce it rather than assuming it.
Do you have any idea what is meant here? (Again, I realise that this is a very minor point, but I would like to understand the paper in its entirety. It's also possible that this confusion on my part points to a hidden subtlety that I'm not seeing).
Thank you very much for shedding light on these remarks.