Incorrect. Double jeopardy is a legal term with a very specific meaning that has nothing to do with punishment. It is only the prohibition against trying an accused person more than once. (Retrials following the declaration of a mistrial don’t violate double jeopardy because in such cases a trial has not been completed; this includes mistrials due to a hung jury).
The concept of double jeopardy extends to punishment because being tried twice for the same crime includes the jeopardy of being punished twice for the same crime.