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Possible typo! "private key" instead of "public key"

The following paragraph under “Digital Signatures using Public Key Cryptography” says:
“This works, but to make it more efficient, instead of signing the entire lengthy message the sender computes a hash (digest) of message and signs that with his public key instead. The receiver can then re-compute the message hash and compare it with the signed hash to ensure that message was not tampered with.”

I believe the sender signs the key with their private key and the receiver will read it with the sender’s public key.

OR maybe add more clarity to the paragraph. So it becomes that the sender signs the digest message with their private key and then appends the signed digest to the message and signs it with the recipient public key.