Understanding 3D objects from images is fundamental to robotics and AR/VR applications. While recent work has made progress in category-level pose estimation, current representations fail to capture the fine-grained semantics needed for reasoning about object parts, functions, and interactions. In this work, we study category-level 3D correspondence in camera space -- predicting, from a single image, 3D locations that remain consistent across instances within a category -- and show that it can emerge without explicit correspondence supervision by learning a shared morphable object prior. To enable research in this direction, we introduce HouseCorr3D, the first large-scale benchmark for monocular category-level 3D correspondence with 178k images across 50 household object categories, 280 unique instances, and 3D keypoint annotations directly on CAD models. Crucially, HouseCorr3D provides amodal correspondence labels for occluded regions and explicit symmetry annotations, addressing key limitations of existing datasets. We further propose Morpheus, a method that learns morphable category-level shape priors by disentangling canonical shape, deformation, and object pose. Through this shared canonical grounding, semantically meaningful 3D correspondences in camera space emerge implicitly. These emerging 3D correspondences set a new state of the art on HouseCorr3D, demonstrating that semantic 3D object understanding can arise without direct correspondence supervision. Data and code are publicly available at https://github.com/GenIntel/HouseCorr3D.