This report is a supplement to the report by the same authors entitled “Can the Mauritius Convention serve as a model for the reform of investor-State arbitration in connection with the introduction of a permanent investment tribunal or an appeal mechanism? Analysis and roadmap” (“CIDS Report”). It analyzes in more depth one aspect already addressed in the CIDS Report, namely the composition of a multilateral investment court (or ITI) and of an appeal mechanism for investor-State awards (or AM). For that purpose, it carries out a comparative analysis of the composition of existing international adjudicatory bodies and in part also arbitral institutions, and seeks to chart the main options for the composition of a prospective ITI and AM. If States were to pursue a reform initiative aimed at institutionalizing investor-State dispute settlement through the creation of these types of adjudicatory bodies, adjudicators would no longer be appointed by disputing parties but would essentially (though nor necessarily exclusively) be chosen by the parties to the instrument establishing the new adjudicatory bodies (section II). This fundamental shift poses a myriad of questions, which the paper seeks to examine.