A 2-categorical extension of the Reshetikhin–Turaev theory

We concretely construct a 2-categorically extended TQFT that extends the Reshetikhin–Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov–Voevodsky 2-vector spaces.