 Solution

Bilateral authentication means that each participant obtains assurance of the other's identity and participation in the relevant run: Alice authenticates Bob and Bob authenticates Alice. An observed name in a message is not such assurance. The protocol must prevent an adversary from supplying the evidence while the named peer is absent.

Session key transport means conveying an already selected session key securely to its intended users. Here Tim selects $k_{AB}$ and sends it inside separate public-key-encrypted tickets for Alice and Bob. This differs from having Alice and Bob jointly derive a key to which both contribute.