Who gave authority to Tywin to appoint Tyrion as Hand of the King?

Question

Before I explain my question, let me state that I have not read the books. I have only seen the show.

So to the question, Tywin is appointed as Hand of the King after he won the Battle of the Blackwater Bay. I believe that is the first time Tywin meets Joffrey (in the show). But before the events of the Blackwater Bay, when Tywin was in battle with Robb Stark, he appoints Tyrion as Hand of the King in his stead. I know that it is just a temporary appointment. But in order to do that, doesn't Tywin need to be already appointed as Hand by the King by Joffrey? In the shows, after Robert is killed, Joffrey just proclaims himself as the King and that's all.

When did Joffrey name Tywin as Hand of the King before Stannis' attack on King's Landing?

Answer 1

Cersei Lannister had appointed Tywin Lannister as the hand of the king. Because Tywin Lannister was busy fighting Starks he decided to nominate Tyrion Lannister as the hand of the king

“This is absurd,” the queen said at last. “My lord father has sent my brother to sit in his place in this council. He bids us accept Tyrion as the Hand of the King, until such time as he himself can join us.”

_{^{A Clash of Kings}}

From the show:

Answer 2

Joffrey is basically a puppet of his mother, and his mother is partly a puppet of her father, Tywin Lannister. Tywin, head of the Lannisters and their region of the Seven Kingdoms, is the one that is really arranging everything. What he says is enforced by threat of steel.

Joffrey becomes king because he is supposedly the blood son of King Robert. Due to his age, the Queen becomes Queen Regent, ruling somewhat in his place. Tywin is named Hand of the King due to his connection to King Joffrey and Queen Regent Cersei. He was also previously the Hand of the Mad King for 20 years and as head of House Lannister, is well suited for the job. Due to needing to lead the army, he is well within his rights to appoint a temporary replacement.

As to when is Tywin named Hand, that happens immediately after Robert is killed Joffrey becomes king and the Lannisters essentially stage a coup. Eddard, the old Hand of King Robert is arrested and replaced.