This would depend on what firmware you wished to update. Most firmware is only covered by their manufacturers official website and using other firmware would invalidate your warranty.
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Just updates your DSi to the latest firmware.
All you have to do is update to the latest firmware.
The function of firmware is to keep all of the hardware working at full capacity. There are usually many updates to firmware after the initial release that would help the hardware to function properly throughout its life.
You need your PSP to be upgraded to the latest firmware, in order to access the PSN and download games.
You may need the latest firmware, upgrade first and try accessing the games after.