I am not the sole copyright holder of the main project.
The main project is under LPGLv2.1 or later. (Phew!) Sorry, I forgot to mention the "or later". Oops!
The AGPLv3 component is not written by me, it is independent. That's why I want to include it.
Thanks for the clarifiers, but there is one final thing that confuses me: I have looked at the charts but they always seem to leave out the AGPLv3. Only GPLv3 is shown.
But yeah, if I can relicense "LGPLv2.1 or later" to AGPLv3 directly, that would be the best. I am pretty sure "LGPLv2.1 or later" can be directly relicensed to GPLv3, I'm just puzzled if that's ALSO working if the goal is AGPLv3 instead. The "A" is important.