If your child misbehaved at daycare, would you punish him at home?

Um, I wouldn't. Your son is young and how is he going to remember why he is being lectured or talked to five hours later? I as a parent would want the person watching my kid to have a talk with him. IMO the talk has to be in the moment when the incident happened and if it happens again on my watch then my son will be pulled aside and talked to.