Extension principle for Floer field theories

Let $\mathcal{C}$ be a quilted $2$-category whose underlying $1$-category has Cerf decompositions. Let $\mathcal{F}$ be a Cerf-compatible partial functor from the objects and simple morphisms of $\operatorname{Bor}_{2+1}$ to those of $\mathcal{C}$, preserving adjunctions and satisfying the quilted naturality axiom. Extension principle for Floer field theories. The partial functor $\mathcal{F}$ has a natural extension to a $2$-functor

$$\operatorname{Bor}_{2+1+1}\to\mathcal{C}.$$

The principle is meant to construct Floer field theories from their data on Cerf generators. Its proof is only outlined in the cited source and relies on the Morse $2$-function result that the bordism bicategory is quilt-generated by Cerf decompositions.