The agreement letter from your PhD advisor in the US and the invitation letter from your supervisor in France are both mandatory documents.
The agreement letter should include that your PhD advisor in the US is agreeing that part of your thesis research will be performed in France under the supervision of your French Advisor. This letter should also highlight your work, explain how a stay in France would be useful to both the lab and your project and how it will strengthen the existing or planned collaboration between your home and host laboratories.
While the invitation letter should also demonstrate why you need to do your research in France and how your mobility is an asset for your research team in France. The invitation letter should mention:
The name of the host institution in France
The name of the research Unit (UMR, UPR, US, etc.). For joint research units, please mention all institutions involved.
Name of the supervisor in France
Former, existing and expected research collaborations, academic cooperation and programs between both institutions (in the US and in France), when applicable.
The letter should explain how the candidate would be integrated to the French research unit as a foreign PhD student.