According to the GDPR, a controller is the natural or legal person, public authority, agency or other body which, alone or jointly with others, determines the purposes and means of the processing of personal data1. A processor is a natural or legal person, public authority, agency or other body which processes personal data on behalf of the controller1. The controller and the processor must enter into a contract or other legal act that sets out the subject-matter and duration of the processing, the nature and purpose of the processing, the type of personal data and categories of data subjects and the obligations and rights of the controller2.
In this scenario, BHealthy is the controller for the personal data of its customers, as it determines the purposes and means of the processing, such as conducting research to decide how to market its new line of sunscreens across Europe. Natural Insight is the processor for the personal data that BHealthy shares with it, as it processes the data on behalf of BHealthy for the purpose of determining the price point for the new sunscreens. However, Natural Insight is also a controller for the same personal data when it uses it for its own purpose of improving its machine learning algorithms, which is not part of the contract or legal act with BHealthy. Therefore, Natural Insight is a controller and a processor for the same personal data, depending on the purpose of the processing3.
Reference:
Art. 4 GDPR - Definitions
Art. 28 GDPR - Processor
Guidelines 07/2020 on the concepts of controller and processor in the GDPR I hope this helps you understand the GDPR and the controller-processor relationship better. If you have any other questions, please feel free to ask me.