Hybrid semiconductor-ferromagnetic insulator heterostructures are interesting due to their tunable electronic transport, self-sustained stray field, and local proximitized magnetic exchange. In this work, we present lattice-matched hybrid epitaxy of semiconductor-ferromagnetic insulator InAs/EuS heterostructures and analyze the atomic-scale structure and their electronic and magnetic characteristics. The Fermi level at the InAs/EuS interface is found to be close to the InAs conduction band and in the band gap of EuS, thus preserving the semiconducting properties. Both neutron and X-ray reflectivity measurements show that the overall ferromagnetic component is mainly localized in the EuS thin film with a suppression of the Eu moment in the EuS layer nearest the InAs and magnetic moments outside the detection limits on the pure InAs side. This work presents a step toward realizing defect-free semiconductor-ferromagnetic insulator epitaxial hybrids for spin-lifted quantum and spintronic applications without external magnetic fields.