In Bar-Natan's On Khovanov's categorification of the Jones polynomial (https://arxiv.org/abs/math/0201043), the claim in section 3.2 when constructing the differential $d_\xi$ is that $d_\xi$ ought to be degree $0$ "so as not to undermine Theorem 1," but Theorem 1 doesn't depend in any way on the maps $d_\xi$. Why should these $d_\xi$ be of degree $0$?
This is important because if we want $d_\xi$ to be obtained from a Frobenius algebra structure on $V$, having a degree $0$ differential gives us an essentially unique choice of multiplication and comultiplication; without this restriction, many Frobenius algebra structures are possible.