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The Climate-Carbon cycle Feedback (CCF) affects emission metric values. In the Fifth Assessment Report (AR5) of the Intergovernmental Panel on Climate Change metric values for Global Warming Potentials (GWP) and Global Temperature Potentials (GTP) are reported both with and without CCF for non-CO2 climate forcers, while CCF is always included for CO2. The estimation of CCF for non-CO2 climate forcers in AR5 is based on a linear feedback analysis. This study compares that approach with an explicit approach that uses a temperature dependent carbon cycle model. The key difference in the CCF results for non-CO2 climate forcers is that, with the approach used in AR5, a fraction of the CO2 signal induced by non-CO2 forcers will persist in the atmosphere basically forever, while, with the approach based on an explicit carbon cycle model, the atmospheric CO2 signal induced by non-CO2 forcers eventually vanishes. The differences in metric values between the two model approaches are within +/- 10% for all well-mixed greenhouse gases when the time horizon is limited to 100 yr or less, for both GWP and GTP. However, for long time horizons, such as 500 yr, metric values are substantially lower with the explicit CCF model than with the linear feedback approach, up to 30% lower for GWP and up to 90% lower for GTP.