Transfer learning (TL) is a well-established machine learning technique to boost the generalization performance on a specific (target) task using information gained from a related (source) task, and it crucially depends on the ability of a network to learn useful features. I will present a recent work that leverages analytical progress in the proportional regime of deep learning theory (i.e. the limit where the size of the training set P and the size of the hidden layers N are taken to infinity keeping their ratio P/N finite) to develop a novel statistical mechanics formalism for TL in Bayesian neural networks. I'll show how such single-instance Franz-Parisi formalism can yield an effective theory for TL in one-hidden-layer fully-connected neural networks. Unlike the (lazy-training) infinite-width limit, where TL is ineffective, in the proportional limit TL occurs due to a renormalized source-target kernel that quantifies their relatedness and determines whether TL is beneficial for generalization.