While thinking about monads in the theory of denotational semantics, I have made an observation about the Kleisli category that I would like to check

Suppose $F : \mathcal D \to \mathcal C$, $G : \mathcal C \to \mathcal D$, is an adjunction, so that $T = GF : \mathcal D \to \mathcal D$ is a monad. There is an operator $ * $ on the Kleisli category that takes $f: X \to TY$ to $f^* : TX \to TY$.

I have an intuition and a sketch proof of the following, but my experience with these things is not strong, so I would like confirmation. In fact, this may just be an obvious or basic result on monads and adjunctions, but I'm not sure. The following statements I *think* I've proved:

$f^* $ is in fact $Gg$ for some $g : FX \to FY$ in $\mathcal C$, and moreover there is a bijection between Kleisli arrows $f : X \to TY$ and morphisms $Gg$ where $g : FX \to FY$. The inverse of the $*$ operation is given by taking $f' : TX \to TY$ to $f' \circ \eta_X$, where $\eta$ is the monad unit.

In a sense this means that the Kleisli category is "as much of $\mathcal C$ as can be represented in $\mathcal D$, given $F$. If what I say is true I suppose it must be something quite elementary, but I don't know where to look. Pointers would be appreciated.