Initiality of the Kleisli adjunction

For a monad $(T,\eta ,\mu )$, the free functor $J:\mathcal{C}\to {\mathcal{C}}_T$ and functor $U:{\mathcal{C}}_T\to \mathcal{C}$ with $UA=TA$ and $U(f)={\mu }_{B}T(f)$ form an adjunction inducing that monad. Every other adjunction inducing the same monad receives the unique Kleisli comparison functor commuting with its left and right adjoints and preserving the adjunction structure. This is initiality among adjunctions inducing the specified monad, with morphisms required to respect that structure.