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There isn't an approval process to subscribe to communities - anyone with a Lemmy account can subscribe. What are you seeing that looks like you need approval?
There can be an approval process for signing up to an instance/server, though.
I think they're referring the Subscribe Pending prompt that some communities give when trying to subscribe.
I've noticed it myself and actually have been wondering the same thing
When I search for a community that isn't on my local instance, I often get a "subscription pending" text, instead of it immediately changing to "joined".
I noticed this too, but it feels more like a pull for the community is scheduled, and usually within minutes I’ll see in under my subscribed.
I don't think it's about approval (there is no "approve subscribers" option when setting up new communities on lemmy). I think it's a bug or burp in the process. I had about 5 pending the other day and now only one shows it now. All of them function normally.
It seems all of them work as well, but if I look at communities I've subscribed to around 20-30 say subscription pending. Most of them are subs I found yesterday, but some are from several days ago.
Here's the bug report for it on github. As far as I can tell it's cosmetic. Some folks report that unsubbing/resubbing fixes it but I haven't tried that.
Edited to add: my sub for this community shows pending. :-)