It's really a matter of how your choose to interpret it. The Ascians would probably say the player character is tempered by Hydaelyn, and it does seem that way from how the primals cannot temper PCs, but it's not explicitly stated. The closest thing to tempering we see done by something that isn't the primals is the effect of Dalamud on Nael van Darnus, which (iirc) also wasn't referred to explicitly as "tempering" (and Dalamud housed an Elder Primal anyway).
