mediatorial kingship

Hi all, I wrote an article with quotes from these men and a few others on the doctrine of mediatorial kingship - for some of them, it's the first time these sections have been translated into English. Unfortunately, when I tried to post it on facebook, their spam filter blacklisted my blog, so...

What does "mediatorial" in "mediatorial kingship of Christ" refer to? What is being mediated? What are the parties on either side? I am interested in mediatorial kingship as it relates to both the Church and the State.