A pretty common problem on early Toyota with the 20R engine - like the Celica, Corona, Truck.
Failure mode is usually when the diaphragm, in the auxiliary accelerator pump, in the carburetor. The AAP is designed to allow a longer accelerator pump stroke when cold.
A failed diaphragm will result in very high CO emission due to sucking gasoline straight through the accelerator pump circuit. First thing I would check would be the manifold and gasket at #4 runner (location of EGR valve) for leaks.
2002 Corolla S, Silverstream, 1.8L 1ZZ-FE (VVT-i)
2003 Matrix XRS, Cosmic Blue, 1.8L 2ZZ-GE (VVTL-i) - RIP
2009 Matrix XRS, Blue Streak, 2.4L 2AZ-FE (VVT-i)