How does the certificate authority generate a signature F?
As far as I understand the question, you ask how a (client-)certificate is issued and generated.

Normally this works like this:


Get an account at the page of the CA. (this is important in later steps)
Validate your identity against the CA, f.ex. using documents like driver-license or passport. Your account will then be listed as "verified as owned by XYZ"
Then you'll create a CSR (certificate-signing-request). This is basically an unsigned X.509 certificate, containing all relevant data about you (=name and your public key)
You send the CSR over to the CA. They will check the your signature on  the CSR to make sure you own the private key associated with the public key. They will also validate that the data in the CSR matches with the data they verified about you.
Now they know that this request is valid. So they'll form a X.509 certificate by taking your CSR and adding their data (where to get CRLs from, validility period, the CA-data,...). Now the certificate is signed by them and the certificate is send back to you. The hash is not stored (as far as I know) and just an intermediate value.


As a formula this would look like this: $Cert=F||S_{CA}(SHA256(F))$.
Where $S_{CA}(X)$ means the signature of the CA on X.

I hope this answers your question. If not use the comments or edit the question.