ScalingStacks

Corollary 5 If $b$ is a flat tangle without closed components then the only grading-preserving endomorphisms of the bimodule ${\mathcal{F}}(b)$ are multiplications by integers, the only grading-preserving automorphisms are $\mathrm{Id}$ and $-\mathrm{Id},$ and ${\mathcal{F}}(b)$ has no endomorphisms of negative degree. If $f:M\to{\mathcal{F}}(b)$ is an isomorphism in $\mathcal{K}_{n}^{m}$ then the only other isomorphism between $M$ and ${\mathcal{F}}(b)$ is $-f.$