Does presidential candidate have to prove citizenship?

If you mean presidents in general, there's no easy answer. Each nation state sets its own rules. Almost universally though the answer will be yes. If you mean the President of the United States of America, then Article 2 of their constitution includes the condition that a presidential candidate must be a "natural born citizen" which means that they must have been born in the United States of America (thus entitling them to citizenship). Each candidate will need to satisfy their own party that they are a fit person to run for president (as, if for no other reason, if they fail on such an obvious point the embarrassment caused would be extremely damaging to them and their political party).