We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg-Moore objects in double categories. We improve upon an earlier result of Fiore-Gambino-Kock in [7] to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category horizontally and vertically, and also in its vertical 2-category. We also prove that a double category admits Eilenberg-Moore objects if and only if a certain parameterized presheaf is representable. Along the way, we develop parameterized presheaves on double categories and prove a double Yoneda Lemma.