Single-vehicle light detection and ranging (LiDAR) has limitations in capturing comprehensive environmental information. The advancement of vehicle-to-infrastructure (V2I) collaboration presents a potent solution to this challenge. During the collaboration, point cloud registration precisely aligns data from various LiDARs, effectively mitigating the constraints associated with data collection by a single-vehicle LiDAR. Registration furnishes autonomous vehicles with a more comprehensive and dependable environmental understanding. Currently, there are various types and performances of LiDAR in practical application scenarios. So, it is more necessary to perform heterogeneous point cloud registration, and there is still a relatively large room for improvement. Consequently, we introduce a coarse-to-fine approach to heterogeneous point cloud registration (C2F-HPCR), establishing the inaugural benchmark for point cloud registration in intricate vehicle-infrastructure collaboration contexts. C2F-HPCR acquires an initial registration matrix through its coarse registration module. Subsequently, it uses the overlap estimation module to extract overlap points between two point clouds. These identified points are inputted into the fine registration module to obtain the final registration matrix. Experiments on the DAIR-V2X-C dataset demonstrate that the recall of C2F-HPCR in heterogeneous point cloud registration is 72.99%. C2F-HPCR shows strong performance in heterogeneous point cloud registration, facilitating efficient registration of vehicle-side point clouds and infrastructure-side point clouds. The code is available at https://github.com/916718212/C2F-HPCR