The appointment of a president or prime minister varies depending on the country's political system.
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Prime Minister: In parliamentary systems, the prime minister is typically appointed by the legislature or parliament. For example, in the United Kingdom, the prime minister is usually the leader of the political party that has the most seats in the House of Commons after an election.
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President: The president can be elected by the legislature or by a direct vote from the public, depending on the country's system of governance. In some parliamentary systems, the president may have a largely ceremonial role and may be appointed by the parliament, while in presidential systems, the president is usually elected directly by the citizens.
If you have a specific country in mind, I can provide more detailed information on how the president or prime minister is appointed in that context.