The assembly of clathrin triskelia into polyhedral cages during endocytosis is regulated by adaptor proteins (APs). We explore how APs achieve this by developing coarse-grained models for clathrin and AP2, employing a Monte Carlo click interaction, to simulate their collective aggregation behavior. The phase diagrams indicate that a crucial role is played by the mechanical properties of the disordered linker segment of AP. We also present a statistical-mechanical theory for the assembly behavior of clathrin, yielding good agreement with our simulations and experimental data from the literature. Adaptor proteins are found to regulate the formation of clathrin coats under certain conditions, but can also suppress the formation of cages.