Eilenberg–Moore algebras in terms of Kleisli ones

Suppose I know what the category of free algebras for a particular monad look like. Can I then describe what the category of Eilenberg–Moore algebras look like? E.g. Suppose that I have a good handle on a category $D$ and that I have two functors $F$ and $G$, with $G:D \to C$ and $F$ left-adjoint to $G$ and $F$ essentially surjective- this guarantees that this adjunction exhibits $D$ as the Kleisli-category for the monad $T:=GF$ on $C$ (see Characterization of Kleisli adjunctions). Is there a way to exhibit the Eilenberg–Moore category $C^T$ as "generalized objects of $D$"? This is somewhat vague of a question I realize, but I'm not sure how to make it more precise. (Feel free to help me do so).

EDIT: To be slightly more specific, in "Toposes, Triples, and Theories" it is said that the Eilenberg-Moore category "is in efect all the possible quotients of objects in Kleisli's category". Can anyone make this precise in a way that answers my question??

One nice result is Street's theorem 14 in The formal theory of monads, generalized in Elementary cosmoi, which says that $C^T$ is isomorphic to the full subcategory of $[(C_T)^{\mathrm{op}}, \mathrm{Set}]$ containing those presheaves that become representable when precomposed with the inclusion $C \to C_T$. That is, $C^T$ is the pullback of $[F^{\mathrm{op}},\mathrm{Set}]$ along the Yoneda embedding. So at least informally you can think of T-algebras as being represented by certain formal colimits of free algebras.