Establishing dense correspondences across image pairs is essential for tasks such as shape reconstruction and robot manipulation. In the challenging setting of matching across different categories, the function of an object, i.e., the effect that an object can cause on other objects, can guide how correspondences should be established. This is because object parts that enable specific functions often share similarities in shape and appearance. We derive the definition of dense functional correspondence based on this observation and propose a weakly-supervised learning paradigm to tackle the prediction task. The main insight behind our approach is that we can leverage vision-language models to pseudo-label multi-view images to obtain functional parts. We then integrate this with dense contrastive learning from pixel correspondences to distill both functional and spatial knowledge into a new model that can establish dense functional correspondence. Further, we curate synthetic and real evaluation datasets as task benchmarks. Our results demonstrate the advantages of our approach over baseline solutions consisting of off-the-shelf self-supervised image representations and grounded vision language models.