If you’ve just enabled Apple Intelligence, it’ll also go through all your Photos and detect objects, faces, pets, POIs, settings, etc… Depending on the amount of photos, this can take a few days to complete. (Should only happen while the phone is connected to a charger, though.) During that time, some sluggishness is to be expected. However, after a few days, the phone should be snappy again.

At least I can’t notice any such issues on my 16 Pro. And I’m using AI since it became available in the Betas here in the UK.