The four beginnings are the moral responses named by Mencius: compassion, shame, deference and the sense of right and wrong. The seven feelings are the ordinary emotions listed in the Book of Rites: joy, anger, sorrow, fear, love, hatred and desire. Zhu Xi's framework holds that everything consists of principle and material force, principle giving a thing its pattern and material force its actuality. The question Korean philosophy spent three centuries on is how the four and the seven map onto that pair.
Yi Hwang's first position was that the four issue from principle and the seven from material force. Gi Dae-seung, a scholar twenty-six years younger, objected by letter that principle cannot issue anything, since in Zhu Xi's system principle does not act and only material force moves. The correspondence ran for eight years and Yi Hwang revised his formulation more than once, arriving at the doctrine of mutual issuance: in the four, principle issues and material force follows; in the seven, material force issues and principle rides upon it.
The stake is not terminological. If moral responses have a different origin from ordinary emotions, then the moral life is grounded in something other than temperament and cannot be reduced to it. Yi Hwang was defending the reality of that difference against a reading he thought would dissolve it. That his junior opponent was answered at length, in writing, over years, rather than dismissed, is part of why the exchange became a model as well as a doctrine.