I proved that $latex (\mathsf{FCD})$ is the lower adjoint of $latex (\mathsf{RLD})_{\mathrm{in}}$.

Also from this follows that $latex (\mathsf{FCD})$ preserves all suprema and $latex (\mathsf{RLD})_{\mathrm{in}}$ preserves all infima.

See Algebraic General Topology and specifically Funcoids and Reloids online article.