This paper emphasizes the critical role of indirect carbon risk exposure transmitted through supply chain linkages in assessing carbon risk and its cross-sectional asset pricing implications. This channel has been overlooked in the burgeoning literature on the carbon risk premium (or greenium). We develop a tractable general equilibrium model that incorporates input-output linkages, carbon emissions, and climate regulatory risks. Analytically, we demonstrate that the carbon risk premium is captured by two sufficient statistics: direct and indirect carbon risk exposure, with the latter representing the network effect. Guided by the model, we empirically measure both direct and indirect exposures and demonstrate a significant cross-sectional carbon risk premium after accounting for firms' indirect carbon risk exposure. Through a decomposition, we quantify that indirect carbon risk exposure accounts for the majority of the premium, surpassing the impact of direct carbon emissions. Furthermore, we provide strong evidence that climate regulatory risks propagating through supply chains help explain the premium.