We show a generic reduction from multiclass differentially private PAC learning to binary private PAC learning. We apply this transformation to a recently proposed binary private PAC learner to obtain a private multiclass learner with sample complexity that has a polynomial dependence on the multiclass Littlestone dimension and a poly-logarithmic dependence on the number of classes. This yields an exponential improvement in the dependence on both parameters over learners from previous work. Our proof extends the notion of $Ψ$-dimension defined in work of Ben-David et al. [JCSS '95] to the online setting and explores its general properties.