Localization has been a challenging task for autonomous navigation. A loop detection algorithm must overcome environmental changes for the place recognition and re-localization of robots. Therefore, deep learning has been extensively studied for the consistent transformation of measurements into localization descriptors. Street view images are easily accessible; however, images are vulnerable to appearance changes. LiDAR can robustly provide precise structural information. However, constructing a point cloud database is expensive, and point clouds exist only in limited places. Different from previous works that train networks to produce shared embedding directly between the 2D image and 3D point cloud, we transform both data into 2.5D depth images for matching. In this work, we propose a novel cross-matching method, called (LC)$^2$, for achieving LiDAR localization without a prior point cloud map. To this end, LiDAR measurements are expressed in the form of range images before matching them to reduce the modality discrepancy. Subsequently, the network is trained to extract localization descriptors from disparity and range images. Next, the best matches are employed as a loop factor in a pose graph. Using public datasets that include multiple sessions in significantly different lighting conditions, we demonstrated that LiDAR-based navigation systems could be optimized from image databases and vice versa.