Imam Mustafa As-Suyuti (d. 1243 AH), may Allah have mercy on him, the great authority of his time from Damascus, said the following on the issue:
If it should occur that both `Eid and Jumu`ah fall on Friday, then the obligation of attending the Jumu`ah prayer is lifted from whoever was present at the `Eid prayer with the Imam earlier on that day.
The reason for this is that the Prophet, peace and blessings of Allah be upon him, performed the `Eid prayer and then said after that, “Whoever wants to come and prayer Jumu`ah thereafter, then let him come.”
Therefore the requirement for attending Jumu`ah on the one that had already attended `Eid is lifted but not the compulsory nature of it. This is due to the fact that if he was present after `Eid at the place where Jumu`ah becomes compulsory, he will then need to prayer Jumu`ah as well. 
Imam `Abdul Ghani Al-Lubadi (d. 1319 AH), may Allah have mercy on him, also spoke on this when he said:
If the `Eid shall fall on a Friday, then the Jumu`ah prayer is not compulsory for the one that prayed the `Eid prayer with the Imam earlier. What is meant is that this individual is not required to be present but the compulsory nature of it still remains.
What this means is that it is not compulsory for him to be present for the Jumu`ah prayer and if he was present when Jumu`ah become compulsory, he would have to prayer the Jumu`ah prayer as well except in the case of the Imam.
The Imam must be present for both Jumu`ah and the `Eid regardless. So if the numbers should be present for the Jumu`ah to be established, he makes the Jumu`ah; but if the numbers are not present, then he is not required to pray Jumu`ah but will instead prayer Zuhr.