That would be fine. The requirements for the FSSC 22000 scheme cover this in ISO 22000:
5.4 Responsibility and authority
Top management shall ensure that responsibilities and authorities are defined and communicated within the organization
5.5 Food safety team leader
Top management shall appoint a food safety team leader who, irrespective of other responsibilities, shall have the responsibility and authority
a) to manage a food safety team (see 7.3.2) and organize its work,
b) to ensure relevant training and education of the food safety team members (see 6.2.1),
6.2 Human resources
The food safety team and the other personnel carrying out activities having an impact on food safety shall be competent and shall have appropriate education, training, skills and experience.
It does not prescribe how responsibilities are communicated it can be as you describe or by job descriptions etc.