Scope of the Khovanov and decorated knot Floer cobordism maps

Apply [F1] to the smooth oriented surface. Its target is ordinary even integral $K{h}^{i,j}$, and identifying a map with its negative removes exactly the one sign ambiguity. Its isotopy and composition conclusions are the first assertion of the Statement.

For the separately decorated surface apply [F2], coloring all w-regions by $U$ and z-regions by $V$. The intermediate multiply based link modules and their endpoint naturality identifications are part of that construction. Zero curvature makes the endpoint objects the specified complexes, and external Theorem A gives filtered equivariant chain-homotopy invariance. This assertion has coefficient field ${\mathbb{F}}_2$ and its own decoration and Spin-c data; no integral Khovanov sign ambiguity or comparison between the two theories is transferred to it.

First require the chosen piece Spin-c structures to have equal restrictions on the intermediate boundary and identify their intermediate complexes by naturality. The sector maps are then composable, and their general composition law is the extension sum in [F2]. If those restrictions differ, the target and source sectors differ and no composite is asserted. On $S^3\times [0,1]$ the Spin-c set has one element, so the extension sum is a singleton for compatible piece structures and empty otherwise. Combining the two independently established assignments proves the entire Statement with its distinct targets, coefficients and equivalence relations. It asserts neither equality of the functors nor a canonical chain-level top generator, and requires no new local reconstruction of the imported Cerf checks.