Is it correct to say that the predicate is the complement of the subject, but the complement of the subject is not necessarily the predicate?

In a sense, it can be said that

In the "subject list" structure, the predicative is used to explain the status of the subject, etc. In a functional sense , we might as well call it "the complement of the subject", although it is called an predicative grammatically due to the structure of the main system.

In addition to the predicative in the main system structure (other Marked by the linking verb be, etc.), other subject complements cannot be called predicatives

For example, He woke up frightened.

frightened is a subject complement, but We can't call it an expression