How did Sweden grant Frederick William sovereignty over Ducal Prussia?

During the First Northern War (1655–1660), King Charles X Gustav of Sweden granted Frederick William, the Elector of Brandenburg, full sovereignty over the Duchy of Prussia in an effort to keep him as an ally.1

Initially, Frederick William had attempted to maintain armed neutrality, but he was eventually forced to exchange Polish suzerainty for Swedish suzerainty and provide military support to the Swedish king. Following the Brandenburg army's performance at the Battle of Warsaw in 1656, the Swedish king formalized this grant of sovereignty.1

It is important to note that this grant did not prevent Frederick William from later negotiating with Poland once Sweden's military position weakened. Ultimately, the end of Polish suzerainty over Ducal Prussia was formally secured through the Treaty of Wehlau in 1657, in which Frederick William agreed to provide troops to Poland against Sweden, and this status was later confirmed by the Treaty of Oliva in 1660.123