Definition of rightly adverb from the Oxford Advanced American Dictionary

      

    rightly

     adverb
    adverb
    NAmE//ˈraɪtli//
     
     
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  1. 1 for a good reason The school was rightly proud of its many successful graduates. He was proud of his beautiful house,and rightly so. Quite rightly, the environment is of great concern.
  2. 2 in a correct or accurate way synonym correctly Rightly or wrongly, many older people are afraid of violence in the streets. As she rightly pointed out, the illness can affect adults as well as children. I can't rightly say what happened. I don't rightly know where he's gone. If I remember rightly, there's a bus at six o'clock. Which Word?right / rightly Right and rightly can both be used as adverbs. In the sense “correctly” or “in the right way,”right is the usual adverb. It is only used after verbs:He did it right. Did I spell your name right?Rightly cannot be used like this. In formal language correctly is used:Is your name spelled correctly? The usual meaning of rightly is “for a good reason” and it is used most often in the phrase “and rightly so”:She disagrees with the policy, and rightly so.It can be used to mean “correctly” before a verb or in particular phrases:As you rightly say, we have a serious problem.
See the Oxford Advanced Learner's Dictionary entry: rightly