It's just that usually when we read articles or books about composting, I never readed description of composting speaking about the action of the psychrophilic organisms. It is composed of a mixed population of organisms. As this heat accumulate and make the temperature go up 40oC or 45oC, mesophilic microorganisms suicide and the thermophilic ones are selected. Usually papers that have studied the evolution of microorganisms during composting, thermophilic actinomycetes are selected during this initial thermophilic phase when the temperature is the hottest because these microorganims are thermotolerant. So they are very important for the composting of this kind of material.
Microorganisms are the principal actors of composting.
I think if you understand what they need or how they work, then you can better understand composting. As I buil the pile, I added a little ammonium nitrate, and water. On day three, temperature was 52oC., and I turned the pile. I saw a lot of white fiberous stuff that was causing the chips to form clumps. I assume this to be fungi that you speak of.