ScalingStacks

Proof. If $F_{1}$ and $F_{2}$ are equivalent as in the lemma, let us show that they are equivalent as in the definition of the skein lasagna module. The only new relation is the isotopy, which can be thought of as a particular instance of (b), where $B_{1}$ is replaced by a slightly smaller ball with the same decoration (and $F_{3}$ is a product cobordism).

Conversely, if $F_{1}$ and $F_{2}$ are equivalent as in the definition of the skein lasagna module, we only have to consider the case when they are related by (b). We can then isotope $B_{i}$ to turn it into the ball $R_{j}$ in the same connected component, and view (b) as a combination of the moves in the lemma. ∎