I have been trying to understand, without success, the definition of "LGP-monoids" on p. 80 of Mochizuki's IUT III and was wondering if anyone could provide some more explanation than what is given there.

For some more details, in the paper (l.13, p.80) these monoids are defined by *pull-back* via a *log-link* of some monoids defined earlier, whose definition I think I understand. What I am having difficulty with is in understanding what *pull-back* via the log-link means since this operation is not defined earlier (as far as I could tell by searching through all the papers in the series) and I could not make any meaningful guesses.

The notation of the paper is very complicated so I do not expect a detailed answer here; a few lines of explanation, or a pointer to some place in the paper where this is made clear, would be very useful.