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We theoretically studied synchronization of chaotic oscillation in semiconductor lasers with chaotic light injection. Feedback-induced chaotic light generated from a master semiconductor laser was injected into a solitary slave semiconductor laser. The slave laser subsequently exhibited synchronized chaotic output for a wide parameter range with strong injection and frequency detuning within the injection-locking regime. Our numerical simulation revealed that the synchronized slave laser exhibits remarkable phase locking, even for chaotic light injection. Consequently, synchronization in phase fluctuations becomes dominant over intensity fluctuations. We found that there exists a parameter range where the slave can synchronize in phase only, with no intensity synchronization. However, synchronization can be completely destroyed, both in phase and in intensity, when the phase locking becomes unstable due to four-wave mixing or excited resonance oscillation. The phase locking was studied analytically and the correspondence between numerical and analytical results was shown. We also analytically examined chaos synchronization based on a linear stability analysis from the viewpoint of modulation response of injection-locked semiconductor lasers to a chaotic light signal. As a result, we verified that such injection-locking-induced chaos synchronization results from a quasilinear response of the bandwidth-broadened slave laser due to strong optical injection.