I am reading "Quantum Invariants of Knots and 3-Manifolds" by Turaev. I have a dificulty to understand the proof of Lemma 2.6 on page 172.

The lemma says that a special ribbon graph drawn on page 167 presents a cylinder. I am sorry that I don't know how to show that ribbon graph here.

I especially don't understand the statement starting "One may check that the cylindrical structures are compatible on the boundary..."

Could you show me the detail and/or an intuitive(geometric) proof?