Hi Jeffro. The letter to Artaxerxes involves objections to rebuilding the city:
(Ezra 4:11-16) . . .“To King Ar·ta·xerx′es from your servants, the men of the region Beyond the River: ... They are rebuilding the rebellious and wicked city, and they are finishing the walls and repairing the foundations. Now let it be known to the king that if this city should be rebuilt and its walls finished, they will not give tax, tribute, or toll, and it will result in a loss to the treasuries of the kings. ... We are making known to the king that if this city is rebuilt and its walls are finished, you will have no control of the region Beyond the River.”
(Ezra 4:17-23) . . .Greetings! And now the official document that you sent us has been clearly read before me. ... Now issue an order for these men to stop work, so that the city may not be rebuilt until I issue an order . . . [Cp. Neh. 1:3; 2:3, 5, 17.]
... Now after the copy of the official document of King Ar·ta·xerx′es had been read before Re′hum and Shim′shai the scribe and their colleagues, they quickly went to Jerusalem to the Jews and used force to stop them.
On the other hand, objections and schemes to halt the rebuilding of the temple were occurring between Cyrus' and Darius' reigns:
(Ezra 4:3-5) . . .“You have no share with us in building a house to our God, for we alone will build it to Jehovah the God of Israel, just as King Cyrus the king of Persia has commanded us.” Then the people of the land were continually discouraging the people of Judah and disheartening them from building. They hired advisers against them to frustrate their plans all the days of King Cyrus of Persia until the reign of King Da·ri′us of Persia.
(Ezra 4:24) . . .It was then that the work on the house of God, which was in Jerusalem, came to a halt; and it remained at a standstill until the second year of the reign of King Da·ri′us of Persia.