We developed a high-power laser system at a wavelength of 399 nm for laser cooling of ytterbium atoms with ultraviolet laser diodes. The system is composed of an external cavity laser diode providing frequency stabilized output at a power of 40 mW and another laser diode for amplifying the laser power up to 220 mW by injection locking. The systematic method for optimization of our injection locking can also be applied to high power light sources at any other wavelengths. Our system does not depend on complex nonlinear frequency-doubling and can be made compact, which will be useful for providing light sources for laser cooling experiments including transportable optical lattice clocks.