Monad induced by an adjunction ($T=GF$)

An adjunction $F⊣G$ with unit $\eta$ and counit $\epsilon$ induces the monadwith unit $\eta$.

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.