The Text

1801 Constitutional Amendment V [Council of appointment, powers construed.]:
And this convention do further, in the name and by the authority of the people of this state, ORDAIN, DETERMINE, AND DECLARE, That by the true construction of the twenty-third article of the Constitution of this state, the right to nominate all officers, other than those who, by the Constitution, are directed to be otherwise appointed, is vested concurrently in the person administering the government of this state for the time being, and in each of the members of the Council of Appointment.

The full provision includes both the amendment and the text it amends – NY Constitution 1777, Article XXIII:
That all officers other than those who, by this Constitution, are directed to be otherwise appointed, shall be appointed in the manner following, to wit: The assembly shall, once in every year, openly nominate and appoint one of the senators from each great district, which senators shall form a council for the appointment of the said officers, of which the governor for the time being, or the lieutenant governor, or the president of the senate (when they shall respectively administer the government), shall be president, and have a casting voice, but no other vote, and, with the advice and consent of the said council, shall appoint all of the said officers; and that a majority of the said council be a quorum; AND FURTHER, The said senators shall not be eligible to the said council for two years successively.