Kleisli comparison functor ($K:{\mathcal{C}}_T\to \mathcal{D}$)

If $L⊣R$ induces a monad $T=RL$, the comparison maps $A$ to $LA$ and maps a Kleisli category arrow $f:A\to TB$ to ${\epsilon }_{LB}L(f)$. It is full and faithful, by the adjunction bijection $\mathcal{D}(LA,LB)\cong \mathcal{C}(A,TB)$. Consequently it is part of an equivalence of categories precisely when every object of $\mathcal{D}$ is isomorphic to some $LA$.