Being deserving of some form of punishment seems, as Mill says, to be part of the very idea of moral guilt and culpability. Similarly, to be strictly considered punishment, a sanction must involve the idea of being deserved by virtue of culpability. We can see this in the emotion of guilt. Consider how you feel and what you feel like doing when you feel guilty. Of course, you feel bad.
Guilt feelings seem to be punitive in their very nature.
C. Nietzsche believes that punishment of this moral kind depends upon projected hostility. The distinctive character of morality cannot be understood apart from the repressed desire to get back at someone. Throughout the greater part of human history punishment was not imposed . . . What is distinctive about moral punishment is the thought that it is deserved, that owing to the wrongdoer s free choice, she deserves to suffer. This suffering serves as a kind of compensation to the victim.