The L 3 photoelectron spectroscopy (PES) spectrum of Co measured in coincidence with the L 3 -VV Auger-electron spectroscopy (AES) peak is analyzed by a many-body theory. The spectrum provides hard evidence that the L 3 PES shakeup/off satellite relaxes fully to the L 3 PES main line prior to the L 3 -hole decay. This is in accord with the experimental finding that except for a very weak AES satellite, there is no appreciable line shape change between the singles (noncoincidence) L 3 -VV AES spectrum of Co measured just above the L 3 PES main-line ionization edge and the one far above the L 1 -level ionization edge. The delocalization time of the valence hole created together with the core-hole in the atomic L shell in Co is shorter than the core-hole lifetime.