How was the correspondence between Anno Domini and Anno Urbis Conditae eras made? How are modern historians proving that this correspondence is correct?
It is much worse than you think. Every kingdom, tribe and city state in the ancient world had its own calendar and year count, and only the rise of large reams caused anything like a uniform dating system over a wide area.
in many societies it was common to date events by the rule of officials. Elected officials might begin and end their terms of office at the new year (which differed in different societies) but monarchs usually began and ended their reigns at different unexpected times during the year, and there were different rules for counting the years of a king's reign in different societies at different times. Using different ways to count the lengths of King's reigns causes the royal chronologies of Israel and Judah in the Bible to not add up, for example.
The Roman consuls usually ruled for exactly one year of the Roman calendar and thus it was usual to date events to the year of the consulship of X and Y. Lists of consuls, fasti, were kept to keep track of the years.
So, for example, during the consulship of Lutatius Catalus and Posthumius Albinus (242 BC) a young Roman aristocrat checking to see when he would be old enough to run for office who knew he was born during the consulship of M. Atilius Regulus and L. Julius Libo (267 BC) could check a list of consuls to see that his 25th birthday would be in the consulship of Lutatius Catalus and Posthumius Albinus.
The Greek historian Ephorus of Cyme (c.400-330 BC) began dating events by Olympiads, four year periods between ancient Olympic games.
The ancient Romans were certain of the day Rome was founded, April 21, the day of the festival sacred to Pales, goddess of shepherds, on which date they celebrated the Par ilia (or Palilia). However, they did not know the exact year the city had been founded; this is one reason they preferred to date their years to the presiding consuls over using the formula A.U.C. or Ab Urbe Condita. Several had been proposed by ancient authorities, and Dionysius of Halicarnassus records these: the Greek historian Timaeus, the first to write a history of the Romans, stated that Rome was founded in the 38th year prior to the first Olympiad, or 814 BC; Quintus Fabius Pictor, the first Roman to write the history of his people, stated Rome was founded in the first year of the eighth Olympiad, or 748/7 BC; Cincius Alimentus claimed Rome was founded in the fourth year of the twelfth Olympiad, or 719/8 BC; and Cato the Elder calculated that Rome was founded 432 years after the Trojan War, which Dionysius stated was the first year of the seventh Olympiad, or 752/3 BC. Dionysius himself provided calculations showing that Rome was founded in 751 BC, starting with the Battle of the Allia, which he dated to the first year of the ninth Olympiad, or 390 BC, then added 119 years to reach the date of the first consuls, Junius Brutus and Tarquinius Collatinus, then added the combined total of the reigns of the Kings of Rome (244 years) to arrive at his own date, 751 BC. Even the official Fasti Capitolini offers its own date, 752 BC.
The most familiar date given for the foundation of Rome, 753 BC, was derived by the Roman antiquarian Titus Pomponius Atticus, and adopted by Varro, having become part of what has come to be known as the Varronian chronology. An anecdote in Plutarch where the astrologer Lucius Tarrutius of Firmum provides an argument based on a non-existent eclipse and other erroneous astronomical details that Rome was founded in 753 BC suggests this had become the most commonly accepted date. Through its use by the third-century writer Censorinus, whose De Die Natali was the ultimate influence of Joseph Justus Scaliger's work to establish a scientific basis of ancient chronology, it became familiar.
So ancient writers estimated that Rome had been founded in 814 BC, 748/7 BC, 719/8 BC, 752/3 BC, 751 BC, and 752 BC, as well as the 753 BC that became the official date.
In the Hellenistic and Roman world, many dating systems were used along with Olympiads, Consular dating, and AUC (Anno Urbis Conditae). They included the Seleucid era (counting from 312 BC), The Pompeian era in Syria (counting from 63 BC), the Spanish era (counting from 38 BC) used in Spain until 1381, the Era of Actium used in Roman Egypt (counting from 31 or 30 BC) and the era of Augustus counting from 27 BC the "foundation" of the Roman Empire).
Starting in ancient times and continuing down through the centuries and millennia, chronologists have tried to find the relationships between different calendars and year counts.
For example they compare king lists. If King X of Country A killed King Y of Country B in an event dated to the fourth year of King X's reign, and the Country B king list shows that King Y of Country B died in the 10th year of his own reign we deduce that King X of Country A became king in the 6th year of the reign of King Y of Country B.
They look for documents that date events in two different dating systems.
If an event was vary famous they look for records from different societies that date it according to different dating systems.
They find the dates recorded for ancient observations of astronomical events like eclipses, oppositions, conjunctions, etc. and compare them with the dates calculated by modern astronomers.
They use radiocarbon dating, dendrochronology and geological dating of earth quakes and volcanic eruptions to try to find scientific dating of historical events.
It is like a gigantic jigsaw puzzle that has not yet been completely put together.
For example, for ancient Egypt and Mesopotamia there are several rival chronological systems that date the earliest historical events centuries apart.
But the chronological system for the ancient near east and mediterranean is pretty accurate and securely established from about 500 BC or so.