What are the foundational sources establishing the legality of judicial authority (Qada)?

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The foundation for judicial authority and its permissibility rests upon the Book (Quran), the Sunnah, and consensus (Ijma). Evidence from the Quran includes the command to judge justly: {O David, indeed We have made you a successor upon the earth, so judge between people in truth and do not follow desire, lest it misdirect you from the way of Allah} [Quran 38:26]. Further support is found in commands such as: {and judge between them by what Allah has revealed} [Quran 5:49], {And when they are called to Allah and His Messenger to judge between them} [Quran 24:48], and affirming faith is conditional upon acceptance of the judge's ruling: {But no, by your Lord, they will not [truly] believe until they make you judge over what they dispute among themselves in their affairs, then find no discomfort within themselves at what you have judged and submit in [full, willing] submission} [Quran 4:65]. The Sunnah is established by the Hadith narrated by Amr ibn al-'As from the Prophet, peace be upon him, stating: "When the judge exerts effort (ijtihad) and is correct, he receives two rewards, and when he exerts effort and errs, he receives one reward," which is agreed upon (Muttafaq 'alayh). Muslims are in consensus regarding the legitimacy of appointing judges and adjudicating between people.