Why do some Church Fathers say that Jesus exterminated the Amalekites?

>“As therefore,” I say, “while Moses was still among men, God took of the spirit which was in Moses and put it on Joshua, even so God was able to cause [the spirit] of Elijah to come upon John; in order that, as Christ at His first coming appeared inglorious, even so the first coming of the spirit, which remained always pure in Elijah like that of Christ, might be perceived to be inglorious. For the Lord said He would wage war against Amalek with concealed hand; and you will not deny that Amalek fell. But if it is said that only in the glorious advent of Christ war will be waged with Amalek, how great will the fulfilment of Scripture be which says, ‘God will wage war against Amalek with concealed hand!’ You can perceive that the concealed power of God was in Christ the crucified, before whom demons, and all the principalities and powers of the earth, tremble.”… A sign of Him that was to be crucified was made… in the type of the extending of the hands of Moses, and of Oshea being named Jesus (Joshua); when you fought against Amalek: concerning which God enjoined that the incident be recorded, **and the name of Jesus laid up in your understandings; saying that this is He who would blot out the memorial of Amalek from under heaven.** Now it is clear that the memorial of Amalek remained after the son of Nave (Nun): but He makes it manifest through Jesus, who was crucified, of whom also those symbols were fore-announcements of all that would happen to Him, the demons would be destroyed, and would dread His name, and that all principalities and kingdoms would fear Him; and that they who believe in Him out of all nations would be shown as God-fearing and peaceful men. > >Justin Martyr >And in the Apocalypse the same exhortation of divine preaching speaks, saying, “Hold fast that which thou hast, lest another take thy crown;” which example of perseverance and persistence is pointed out in Exodus, when Moses, for the overthrow of Amalek, who bore the type of the devil, raised up his open hands in the sign and sacrament of the cross, and could not conquer his adversary unless when he had stedfastly persevered in the sign with hands continually lifted up. “And it came to pass,” says he, “when Moses raised up his hands, Israel prevailed; but when he let down his hands, Amalek grew mighty. And they took a stone and placed it under him, and he sate thereon. And Aaron and Hur held up his hands on the one side and on the other side, and Moses’ hands were made steady even to the going down of the sun. **And Jesus routed Amalek and all his people.** And the Lord said unto Moses, Write this, and let it be a memorial in a book, and tell it in the ears of Jesus; because in destroying I will destroy the remembrance of Amalek from under heaven.” > >Cyprian