Carbon fixing takes place via photosynthesis plus the light independent Calvin cycle. Hence the extensive greening that's taken place since CO2 elevation. This was first noticed via satellite in early 1980s. Overall there has been an increase of around 25-50% of vegetated land area. Productivity gains have been around 10%. Elevated CO2 has been the main driver. Plants got to get their (45-50% of their mass) carbon from somewhere.