Not strictly correct: the wording in Law 05 is
* a player is injured as the result of a physical offence for which the opponent is cautioned or sent off (e.g. reckless or serious foul challenge), if the assessment/treatment is completed quickly
I think the general consensus is that if the player is still receiving treatment once the caution has been administered then it's not necessarily going to be considered as being dealt with quickly. Perhaps Mr Friend thought it had taken too long. Very open to interpretation as how quickly is quickly. I assume there was no blood? That would be another reason, if there was blood he'd still have to go off and get that seen to.