Asymmetric Encryption + Signature workflow

Let´s say that Alice wants to send an encrypted message that only Bob can read, plus confirm that Alice is the person that wrote that message.

What is the procedure that will take place?

When I see digital signature workflow I always see that the message it´s transferred “clear” to the receiver (so Bob can hash the document in his side and compare it with the hash received by Alice), but the encrypted part (using Alice´s private key) is always the hash.

What happens when we want to encrypt also the message to a specific person? There will be a “second encryption” process, in which the message will be encrypted by Bob´s public key?

Just an example to confirm:

  • Alice produce digital signature: creates a hash of the document + encrypt it using her private key.

  • Alice encrypts the message itself: Takes the document and encrypt it using Bob´s public key (so only he can decrypt it, using his private key). (I guess that if the document is big can take a long time to finish the process, right?)