The EU Emission Trading System (ETS) is the oldest and currently the largest carbon market in the world, but its purpose of stimulating carbon emissions via trading profits remains unexamined. Based on the complete firm-level transaction records of the EU ETS Phases I and II, here we show that the participating firms' trading profits and their emission abatements are positively correlated, and the correlation becomes stronger in Phase II than Phase I. Specifically, we observe that non-linearity exists in the correlation; higher firm-level emission abatements can realize larger trading profits. This pattern affects the market fairness, though it may be helpful to incentivise emission abatements. The correlation is more regulated in Phase II than it is in Phase I, thereby indicating that the Phase II is more mature. We also observe that the state-level abatements are largely driven by industrial giants.