en.wikipedia.org/wiki/Pope_Innocent_X

This is a chronology error. Innocent IX reigned only in 1591, but the Pamphilj family’s Roman branch was established much earlier.

• Treccani says the Roman branch began when Antonio Pamphili entered the Curia in the second half of the 15th century, obtained the office of procuratore fiscale in 1460, and settled in Piazza Pasquino in 1479.
• A Cambridge University Press article states that "A branch of the family transferred to Rome during the reign of Sixtus IV (1471-84), whose successor, Innocent VIII (1484-92), promoted Andrea Pamphilj" and adds that Innocent X’s papal name acknowledged the family’s rise under Innocent VIII.

So the family did not come to Rome during the pontificate of Innocent IX.

The appointment date here is off by about two years.

• Treccani states that after Gregory XV became pope in February 1621, "Il 26 marzo con un breve fu nominato nunzio a Napoli" — he was appointed nuncio to Naples on 26 March 1621.
• Catholic-Hierarchy likewise lists him as appointed Apostolic Nuncio to Naples on 26 Mar 1621.
• The Doria Pamphilj family biography also says he served as Papal Nuncio to Naples from March 1621 to March 1625.

So 1623 is incorrect; the correct year is 1621.

This misstates Francesco Barberini’s diplomatic role. Contemporary scholarship and modern reference works describe Barberini’s 1625–1626 missions as a legation, not a nunciature.

• Treccani says Urban VIII recalled Pamphilj in March 1625 to assign him to the service of Cardinal Francesco Barberini, who had been "nominato legato a latere in Francia"; it then says Barberini was again "nominato legato a latere in Spagna" in 1626.
• A scholarly study on papal finances and diplomacy is explicitly titled "le cas de la légation du cardinal Francesco Barberini en France en 1625".
• Another scholarly study on Pamphilj’s Spanish correspondence says he accompanied Francesco Barberini "nella sua legazione in Francia e in Spagna" as datary.

So Barberini was acting as a papal legate, not a nuncio.

This identifies the wrong institution. Treccani’s biographical entry says Giovanni Battista Pamphilj earned his doctorate in civil and canon law at the Sapienza of Rome. A modern biographical page associated with Palazzo Pamphilj likewise states that he graduated in law from the University of Sapienza in 1594.

The Collegio Romano and La Sapienza were different institutions, so saying he graduated from the Collegio Romano is inaccurate.