The paper analyses the reasoning provided in the judgment of the Court of Justice of the European Union of 30 March 2023, C-34/21 (Hauptpersonalrat der Lehrerinnen und Lehrer beim Hessischen Kultusministerium), which is the first ruling of this court containing an interpretation of Article 88(1) and (2) of GDPR, being the so-called "data processing agreement clause" addressed to Member States in relation to the processing of personal data in the context of employment. The author's considerations cover both the personal and material scope analysed by the CJEU and, primarily, an assessment of the proposed line of interpretation of the concept of "more specific rules" that may be adopted by Member States.