ct.category theory - Characterization of Kleisli adjunctions

There is a unique functor $mathbf{Kl}(GF) rightarrow mathbf{D}$ commuting with the adjunctions from $mathbf{C}$, since the Kleisli category is initial among adjunctions inducing the given monad; and this functor is always full and faithful, since $mathbf{Kl}(GF)(A,B) cong mathbf{C}(A,GFB) cong mathbf{D}(FA,FB)$.

So this functor will be an equivalence iff it is essentially surjective, and an isomorphism iff it is bijective on objects. But its object map is just the object map of $F$.

So $mathbf{Kl}(FG)$ is equivalent to $mathbf{D}$ compatibly with the adjunctions from $mathbf{C}$ precisely when $F$ is essentially surjective, and isomorphic just when $F$ is bijective on objects.