We present a universal prior model for 3D head avatar with hair compositionality. Existing approaches for building generalizable prior for 3D head avatar often model face and hair in a monolithic manner, where the inherent compositonality of the human head and hair is not considered. It is especially challenging for the monolithic model to self-discover the compositionality of face and hair when the dataset is not large enough. Moreover, extending the monolithic model for applications like swapping faces or hairstyles in 3D is not straightforward. Our prior model explicitly accounts for the compositionality of face and hair, learning their priors separately. To learn a disentangled latent spaces of face and hair of 3D head avatars, we propose a synthetic hairless data creation pipeline for dehairing the studio-captured dataset with estimated hairless geometry and hairless texture obtained from diffusion prior. Using a paired dataset of hair and hairless captures, disentangled prior models for face and hair can be trained by leveraging compositionality as an inductive bias to achieve disentanglement. Our model's inherent compositionality enables a seamless transfer of face and hair components between avatars while maintaining the subject's identity. Furthermore, we demonstrate that our model can be finetuned with a monocular capture to create hair-compositional 3D head avatars for unseen subjects, highlighting the practical applicability of our prior model in real-world scenarios.