How do new claims and their signatures get generated from a collection of existing VCs

[r-soltani]

My understanding of verifiable claim (or more specifically verifiable credentials) is that it is made of many components including a 'Proof' attribute that describes the signature of the claims.

When an identity holder is required to share one or a number of the claims that are in his possession with a Verifier, how does the signature attaches to the new verifiable claim (or I guess the correct term is verifiable profile) is calculated on the spot? Does this new signature require the original key material of the issuer?

Is the VC standard assuming the use of approaches such as Idemix to generate the new signature for the verifiable claims passed from owner to verifier?