The Khovanov Frobenius TQFT respects the Bar-Natan relations

By the assumed functoriality of $F$, gluing cobordisms is sent to composition of maps and disjoint union is sent to tensor product. Thus the only additional relations to check before factoring through the quotient are the three local relations in [F2].

A sphere is evaluated by $ϵ\eta$, and $ϵ(\eta (1))=ϵ(1)=0$. A torus is evaluated by $ϵm\mathrm{\Delta }\eta$; on the generator $1$, $\mathrm{\Delta }(1)=1\otimes X+X\otimes 1$, then $m\mathrm{\Delta }(1)=2X$, and $ϵ(2X)=2$. These are exactly the S and T relations, including when glued into any surrounding cobordism because the TQFT respects composition and tensor product.

In the four-tube relation, each side consists locally of two ways to attach one tube. Applying the TQFT to the local four-disk input $1$, either side gives

This follows by applying $\mathrm{\Delta }(1)=1\otimes X+X\otimes 1$ at each tube attachment and $ϵ(1)=0,ϵ(X)=1$ at the two cut disks. The equality on the local input implies equality as maps: tensoring with identity maps and composing with the maps for the unchanged part of the surface preserves equality. Thus the four-tube relation is respected.

Every generating local relation of ${\mathsf{C}\mathsf{o}\mathsf{b}}_{BN}$ is respected by steps 1.2 and 1.3, while the Frobenius identities respect the isotopy and gluing presentation by step 1.1. Therefore the TQFT factors through the quotient defining ${\mathsf{C}\mathsf{o}\mathsf{b}}_{BN}$, as asserted.