ScalingStacks

Theorem 18. Let $(\mathcal{C},\otimes,I,T,\tilde{T}_{(-|-,-)},\tilde{T}_{(-,-|-)})$ be a semistrict braided monoidal 2-category, and let $\mathcal{Z}(\mathcal{C})$ be its center. Then there is a braided monoidal 2-functor $\mathcal{F}:\mathcal{C}\to\mathcal{Z}(\mathcal{C})$ given as follows:

	$\displaystyle\mathcal{F}(A)$	$\displaystyle=(A,T_{A,-},\tilde{T}_{(A|-,-)})$
	$\displaystyle\mathcal{F}(f)$	$\displaystyle=(f,T_{f,-})$
	$\displaystyle\mathcal{F}(\alpha)$	$\displaystyle=\alpha$

Moreover $\mathcal{F}$ is injective on objects, morphisms and 2-morphisms, and surjective on 2-morphisms.