We introduce LOKI, a compute-efficient framework for co-designing morphologies and control policies that generalize across unseen tasks. Inspired by biological adaptation -- where animals quickly adjust to morphological changes -- our method overcomes the inefficiencies of traditional evolutionary and quality-diversity algorithms. We propose learning convergent functions: shared control policies trained across clusters of morphologically similar designs in a learned latent space, drastically reducing the training cost per design. Simultaneously, we promote divergent forms by replacing mutation with dynamic local search, enabling broader exploration and preventing premature convergence. The policy reuse allows us to explore 780$\times$ more designs using 78% fewer simulation steps and 40% less compute per design. Local competition paired with a broader search results in a diverse set of high-performing final morphologies. Using the UNIMAL design space and a flat-terrain locomotion task, LOKI discovers a rich variety of designs -- ranging from quadrupeds to crabs, bipedals, and spinners -- far more diverse than those produced by prior work. These morphologies also transfer better to unseen downstream tasks in agility, stability, and manipulation domains (e.g., 2$\times$ higher reward on bump and push box incline tasks). Overall, our approach produces designs that are both diverse and adaptable, with substantially greater sample efficiency than existing co-design methods. (Project website: https://loki-codesign.github.io/)