The specialised agency of the United Nations, which was also an essential wing of the League of Nations, is the International Labour Organisation (ILO).
The League of Nations was established after World War I to promote peace and cooperation between countries. One of its key components was the International Labour Organisation, founded in 1919, aimed at promoting social justice and internationally recognized human and labour rights.
When the United Nations was founded in 1945, following World War II, the ILO became the first specialised agency within the UN. Its role and principles remained consistent with its original mission: advancing social and economic justice through setting international labour standards.
None of the other options, such as the World Health Organisation, World Trade Organisation, or Food and Agriculture Organisation, were part of the League of Nations.