Big puzzle, but whatever it was, they shied off when they saw it. More than one possible set of circumstances could apply; maybe what they were is deliberately unimportant. What is known is that no accusers were left for a guilty sinner who was pardoned and warned from future sin.
Jesus, being the fulfillment of the law bodily, did adjudicate the law in this case...but not after the order of Aaron and the levites but after the order of Melchizidek. The priestly order of Aaron would break the bruised reeds and quench the faint embers as was its function but Melchizidek doesn’t ignore the law, but rather allows for restoration under grace for those who are repenting. Jesus was of the priestly order of Melchizidek and could judge with righteous judgment, and not ‘as a man sees’.
As God, Christ had the authority to cast the first stone, and as Revelation points out he will do so as the coming “Faithful and True”! Yet, for the prostitute, he was simply her Redeemer and Restorer!