  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 our earlier
result in "Monads in Double Categories", JPAA 215:6, pages 1174-1197, 2011, 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. 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-categorical Yoneda Lemma.

Category Theory