I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.

It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;

The function $(M, \partial_{-}M, \partial_{+}M) \mapsto \tau(M): T(\partial_{-}M) \to T(\partial_{+}M)$ extends the modular functor $T$ to a non-degenerate topological quantum field theory.

The proof of the functoriality is unclear for me. I tried to look at Turaev's papers but its harder to understand.

Could you suggest me a textbook or paper etc that explains this theorem or similar material?

Or could you show me more detailed proof of functoriality here?