Here we derive and experimentally test a free energy functional that captures the adhesion of DNA-coated emulsion droplets. Generalizing previous approaches, the theory combines important energetic and entropic effects of microscopic DNA mechanics and droplet elasticity. It simultaneously predicts adhesion size, morphology, and binder concentration as a function of experimental control parameters. Notably, droplets transition from undeformed binding to flat droplet interfaces at a characteristic DNA coverage. These equilibrium predictions agree quantitatively with experiments on droplet-substrate and droplet-droplet binding, revealing a weak effective binding strength of $3.7\pm0.3\text{k}_{\text{B}}\text{T}$ owing to entropic costs. Our results open the path to rich design strategies for making colloidal architectures.
