I’ve just proved the following:
Theorem $latex (\mathsf{FCD}) (\mathsf{RLD})_{\Gamma} f = f$ for every funcoid $latex f$.
For a proof see this online article.
I’ve also posed the conjecture:
Conjecture $latex (\mathsf{FCD}) : \mathsf{RLD} (A ; B) \rightarrow \mathsf{FCD} (A ; B)$ is the upper adjoint of $latex (\mathsf{RLD})_{\Gamma} : \mathsf{FCD} (A ; B) \rightarrow \mathsf{RLD} (A ; B)$ for every sets $latex A$, $latex B$.
I have also proved that $latex (\mathsf{RLD})_{\Gamma}$ is neither upper nor lower adjoint of $latex (\mathsf{FCD})$ (see the same online article above).