They coordinated to pick the person who was the best political and spiritual fit.
How the West European monarchs and popes relate to the selection of bishops in the Middle Ages?
While the majority of countries worldwide and in Europe are republics (have a head of state who is directly or indirectly elected), the European Union still has seven monarchies, whose heads of state (a monarch) inherit their positions and often hold them for life or until they abdicate.
The majority of monarchies in Europe are constitutional monarchies, which means that neither the monarch, who is either legally prohibited from doing so, nor the monarch, who does not make use of the political powers bestowed upon the office by convention, has any influence over the politics of the state.
Bishops were initially chosen by the local clergy with the blessing of the bishops in the area. The bishops who oversaw the election and carried out the ordination "put a freshly elected bishop in office and granted his authority."