Mishra took charge as Rajashtan Governor on September 9, after being shifted from Himachal Pradesh where he was appointed as Governor on July 22.
A veteran leader from Uttar Pradesh, he has been a state and Union Minister, as well as serving as head of the party's Uttar Pradesh unit and as national Vice President.
Koshyari has served as Chief Minister of Uttarakhand and a member of Rajya Sabha.
The Congress has raised question on Koshyari's conduct with party spokesperson Manish Tewari said that "the question is not that truth has won... the larger question today is and we do hope that the Hon'ble Supreme Court will go into that the manner in which the Constitution, rules, regulations, precedents, and traditions were completely thrown to the winds when the Governor acted in the most partisan manner possible".
Arguing in the Supreme Court on the petition filed by the Shiv Sena-NCP-Congress combine, senior Congress leader and lawyer Kapil Sibal had charged the Governor with acting in a biased and mala fide manner and contrary to the law established by the top court.
A Congress leader said that Governor's position is untenable and the President should immediately replace him after the court verdict.
Governor Koshyari on early Saturday invited Devendra Fadnavis to form the government and swore him in as Chief Minister and Ajit Pawar as Deputy Chief Minister at 8 a.m. Prior to the oath-taking, the Central government, on the Governor's recommendation, lifted President's rule from the state after getting the President's consent.
( With inputs from IANS )