Rightly

adv.thoughtful, kindprofessional

Meaning 1

Correctly, accurately, or with sound judgment.

Reach for it when confirming that an understanding or action accords with the facts

The inspector rightly identified moisture as the main cause.

everyday
properly
when following an accepted method is intended
not when Properly can enforce convention rather than truth.
professional
accurately
when representation or measurement matches reality
not when Accuracy may be exact without being wise.
professional
correctly
when factual or procedural accuracy is central
not when Correctly lacks the judgment or moral shade of rightly.
formal
soundly
when judgment rests on good reasons
not when Soundly does not claim a uniquely correct answer.

Justly or for good moral reason.

Reach for it when affirming that praise, concern, or treatment is deserved

The volunteers were rightly proud of the safe crossing.

everyday
fairly
when impartial treatment is the point
not when Fairly can also mean moderately.
warm
deservedly
when praise, blame, or reward has been earned
not when Deservedly does not cover a morally correct action itself.
professional
appropriately
when the response fits the circumstances
not when Appropriate conduct can still be morally questionable.
formal
justly
when fair and deserved treatment is precise
not when Justly is narrower and more legal or moral.
formal
legitimately
when a claim or authority is valid
not when Legitimacy can follow law without moral rightness.

rightly conclude
Reviewers rightly conclude that another test is needed.
rightly praised
The maintenance team was rightly praised for preventing the failure.
quite rightly
Residents quite rightly asked who would verify the figures.
rightly point out
The apprentice rightly pointed out the missing guard.

The definition and the example on this page are written here. The dictionaries above ground which meaning is being described, not the wording.