Using large-scale ab initio calculations and taking the two-dimensional C2N monolayer as a substrate, we sampled a large combinatorial space of C2N-supported homonuclear and heteronuclear dual-atom catalysts and built a detailed view of catalytic activity and stability toward the oxygen reduction reaction (ORR) and oxygen evolution reaction (OER). The results indicate that regulating combinations of metal pairs could widely tune the catalytic performance. Pd2-, Pt2-, and PdPt-C2N could effectively balance the adsorption strength of intermediates and achieve optimal bifunctional activity. The favorable catalytic performance could also be realized on GaPd-C2N for the ORR and PdRh-C2N for the OER, surpassing corresponding homonuclear counterparts. The thermodynamic and electrochemical stability simulations reveal that these metal pairs can be stably anchored onto the C2N matrix. Multiple-level descriptors, including Gibbs free energy, d-band center, and bonding/antibonding orbital population, are established to track the activity trend and reveal the origin of activity, indicating that catalytic activity is intrinsically governed by the d-band center of metal pairs.