Synchronization among coupled oscillators is a common feature of symmetrically coupled networks with homogeneous, i.e., identical, oscillators. Recently, it was reported [T. Nishikawa and A. Motter, Phys. Rev. Lett. 117, 114101 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.114101 and Y. Zhang, T. Nishikawa, and A. E. Motter, Phys. Rev. E 95, 062215 (2017)2470-004510.1103/PhysRevE.95.062215], however, that in networks with asymmetrically coupled oscillators, synchronization can only be found to be stable when the oscillators are heterogenous or nonidentical. In this manuscript, it is proven, mathematically, that the conclusions in those works are incorrect, and that stable synchronization states can, and do, exist in asymmetrically coupled homogeneous oscillators. Theoretical results are confirmed with numerical simulations.