Ezra 5:13

But in the first year of Cyrus the king of Babylon the same king Cyrus made a decree to build this house of God.

In the first year of Cyrus the king of Babylon - i:e., the first year in which he began to reign in Babylon. Cyrus styles himself, in his famous proclamation, king of Persia (see the note at Ezra 1:1). But it was the Jewish exiles in and around Babylon who went in the first caravan to Jerusalem after the promulagation of the edict, and as it was published shortly after the Medo-Persian conquest of Babylonia, these Jews thought and spoke of Cyrus very naturally as king of Babylon. Accordingly, it will be observed that it was the elders of these first emigrants who spoke of Cyrus by a name which was familiar to them.

The same king Cyrus. The Jews were perfectly warranted, according to the principles of the Persian government, to proceed with the building in virtue of Cyrus' edict. For everywhere a public decree is considered as remaining in force until it is revoked; but the 'laws of the Medes and Persians changed not.'