§ 600-a. Jail time records and certificates. A record shall be kept by the sheriff, or in counties within the city of New York by the commissioner of correction of such city, of all jail time to which the defendant is entitled under subdivision three of section 70.30 of the penal law. In any case where the sheriff or the commissioner of correction of the city of New York has the duty of delivering a defendant to an institution not under his jurisdiction pursuant to sentence and commitment, such person shall deliver a certified transcript of such record to the person to whom the defendant is to be delivered.