John Chrysostom

By His own city here he means Capernaum. For that which gave Him birth was Bethlehem; that which brought Him up, Nazareth; that which had Him continually inhabiting it, Capernaum.

Homily on the Gospel of Matthew 29 ↗

Theophylact of Ohrid

"And He entered into a boat, and passed over, and came into His own city. And, behold, they brought to Him a paralytic, lying on a bed." "His own city" means Capernaum, for it was there that He was living. He was born in Bethlehem, raised in Nazareth, and lived for an extended length of time in Capernaum. This paralytic is not the same as the one mentioned in John (Jn. 5:2-9), for that one was beside the Sheep's Pool in Jerusalem, while this one was in Capernaum. …

Commentary on Matthew ↗

Augustine of Hippo

That Matthew here speaks of his own city, and Mark calls it Capharnaum, would be more difficult to be reconciled if Matthew had expressed it Nazareth. But as it is, all Galilee might be called Christ's city, because Nazareth was in Galilee; just as all the Roman empire, divided into many states, was still called the Roman city. …

Catena Aurea by Aquinas ↗

Jerome

And Jesus got into a boat and crossed over, and came to his own city. And behold, they brought to him a paralytic lying on a bed. And Jesus seeing their faith, said to the paralytic: Take heart, my son, your sins are forgiven. By his city, we understand none other than Nazareth, from where he is also called the Nazarene. And they brought him, as we said before, a paralytic lying on a bed, because he himself was not able to enter. …

Commentary on Matthew ↗

Hilary of Poitiers

Mystically; When driven out of Judæa, He returns into His own city; the city of God is the people of the faithful; into this He entered by a boat, that is, the Church.

Catena Aurea by Aquinas ↗