All once-extended 3D TQFTs are Reshetikhin--Turaev theories

We present a direct geometric construction which extends the Reshetikhin–Turaev TQFT to circles. We prove that this agrees with the generators-and-relations approach of Bartlett–Douglas–Schommer-Pries–Vicary, thus providing an alternative to the Cerf-theoretic part of their classification of once-extended 3-dimensional TQFTs, and deduce as a result that all once-extended 3-dimensional TQFTs arise via our construction.