We made an attempt at UV-completing a class of composite Higgs models based on partial compositeness in four-dimensional spacetime. These composite Higgs models feature a set of hyperfermions charged under a new confining gauge group called hypercolor. Partial compositeness is realized via four-fermion operators composed of one SM fermion and three hyperfermions. Focusing on the third family alone, we find it is possible to construct an anomaly-free UV completion of a composite Higgs model based on Sp(4) hypercolor group by partially unifying the SM color and hypercolor groups, which has the potential to explain the mass hierarchies inside the third family (including the neutrino). With a nontrivial extension we find the model also has the potential to generate masses for the first and second family SM fermions. However, suppression of unwanted flavor and CP violation remains the main open issue.