What is the intention required for one who joins the Imam during Jumu'ah prayer and misses part of it?

Primary text

For one who overtakes the Imam during the Jumu'ah prayer and does not complete what is required for the Jumu'ah prayer to be valid for them, the ruling according to Kharaqi is that they must intend Dhuhr prayer. If they intend Jumu'ah, it is not valid based on the apparent meaning of his statement, as the basis for continuing the prayer requires the initial entry to be with the intention of Dhuhr. This is supported by the implication that if entry is with the intention of Jumu'ah, one cannot build upon it as Dhuhr. Furthermore, the statement attributed to Ahmad, via the narration of Salih and Ibn Mansur, suggests this, as he stated regarding one who starts with the Imam but misses bowing and prostration until the Imam concludes the prayer: 'He starts over with four rak'ahs of Dhuhr.' This suggests commencing the prayer anew because Dhuhr cannot be initiated with the intention of Jumu'ah, nor can it be continued by extension, similar to the relationship between Dhuhr and Asr.