Good compost is not just rotted straw. It is food that Agaricus bisporus can use and most of its competitors cannot. That selectivity is built in Phase II, where thermophilic organisms take up the free ammonia and lock it into their own bodies as microbial protein.
DMR keeps the mass at 45 to 52 degrees Celsius for that conversion, and warns against conditioning below 40 degrees because mesophilic fungi set in and make the compost unsuitable. The conditioning band is not comfort; it is the range where useful organisms beat the others.