An X-ray source via laser-Compton scattering has the advantage of small source, energy tunability and quasimonochromaticity and is expected to be applied in a wide range of fields such as the industry and medical care. In laser-Compton scattering, the luminosity, which represents the collision frequency between the electrons and the photons, is very important. Increasing the luminosity is strongly required for increasing the scattered photon flux. One way to increase the luminosity is tilting electron bunches at the collision point, which is called crab crossing. It is the way to create the head-on collision artificially. The purpose of this study is the proof-of-principle of crab crossing laser-Compton scattering. In this conference, we will report the design optimization and construction of the laser system for the collision and future prospects.