A double doctoral degree (“cotutelle” in French) corresponds to the delivery of two diplomas by two universities for the same thesis on the basis of a single defence. The work is overseen by two thesis supervisors, one in each institution. The diploma must bear mention of the cotutelle between the two universities.
The regulation and procedure to follow are detailed in the Directive pour la mise en place d’une cotutelle internationale de thèse (English version also available) : https://memento.unige.ch/doc/0327/. You will find on the same page the Cooperation Agreement templates (in English and French) to use, as well as the Cotutelle Application Form.
The preparation of an international cotutelle implies the establishment of an Agreement during the first four semesters of the thesis regardless of where it began.