Question: What do we do if Eid falls on a Friday?

Answer: Zaid bin Arqam was asked if two Eid’s ever fell on the same day. He meant if Eid fell on a Friday during the life of the prophet (PBUH). He responded, “Yes” Then he asked, “So what did he do?” Zaid said, “After the prayer he would tell those present that they are permitted to go to the Jumuah prayer later or not as they like.” (Ibn Majah 1089)

Abu Hurairah narrates the Prophet (PBUH) said, “Indeed two Eids have come on the same day today. So whoever would take the permission then the Eid prayer will suffice for Jumuah. We will do both.” (Abu Dawood 1073)

The scholars explain these and other Hadiths narrated by other companions with similar meanings that it would become difficult to celebrate two Eids in one day so the prophet relieved the obligation of Jumuah for those who attend Eid. The safer opinion is after that they would pray Thuhr Salat at home.

As the second narration mentions the prophet (PBUH) would do both. This is not because its better or more virtuous, rather some people may not have been able to attend the Eid prayer and they still have the obligation of Jumuah.

The prophet (PBUH) said “Indeed God loves that you take His permissions just as he loves you take his heavy obligations” (Ibn Hibban 354)

For a more detailed look into the opinions of 4 schools of thought… https://muslimmatters.org/2009/11/26/the-fiqh-ruling-on-jumu%E2%80%99ah-salat-if-eid-falls-on-friday/