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Hello and thank you for your question--it is a good one.
There are varying degrees of amblyopia severity. I suspect that because eye doctors have always prescribed a balanced prescription in the right eye to go with the left, it was because your vision is poor enough in the right eye that even correcting the refractive error in that right eye wouldn't make much difference visually, but may make you uncomfortable walking or performing tasks of daily living.
Depending on how dense or visually significant your cataracts are, you may elect to just do surgery in the left eye only, in the right eye only to see how it helps, in both eyes, or not at all. If your cataracts are advanced and you are having significant problems with distance vision, near vision, needing more light to be able to read, and glare (bright headlights on cars at night or bright sunlight during the day causing your vision to get worse), then you should seriously think about doing the left eye at least.
As far as finding a surgeon who can perform this surgery with respect to your amblyopic status, there are many ophthalmic surgeons who can do this and have experience in dealing with amblyopia in one eye and cataracts in both eyes. There isn't a specific subspecialty specifically that would combine these two expertises together so much; a pediatric ophthalmologist deals with amblyopia on a consistent basis and cataract surgeons see it also, but it is not part of any of their subspecialty training. Something you may consider is seeing a comprehensive ophthalmologist who sees both children and adults and that surgeon is going to have experience treating amblyopia. However, any comprehensive ophthalmologist should know how to treat this situation.
My best recommendation would be to get recommendations from many people regarding a particular ophthalmic surgeon and then trust your gut. If you like and trust your doctor, those are two of the biggest things to trust.
Does that help answer your questions?
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My opinion is solely informative and does not constitute a formal medical opinion or recommendation. For a formal medical opinion and/or recommendation you must see an eye doctor.