ScalingStacks

Proof. Let $L_{1}\in B_{1}$ and $L_{2}\in B_{2}$ and write $L$ for the resulting split disjoint union in $B\stackrel{{\scriptstyle\mathrm{def}}}{{=}}B_{1}\#_{\partial}B_{2}$. We can find a diffeomorphism $\phi\colon B\to{\mathbb{R}}^{3}$ such that not only is $L$ generic and blackboard-framed, but also the $z$-projections of the $L_{1}$ and $L_{2}$ components of $L$ are contained in disjoint disks in ${\mathbb{R}}^{2}$. Then, monoidality on the chain level is manifest in the definition of $\mathrm{KhR}_{N}$, and we get

	$\displaystyle\mathrm{KhR}_{N}(B_{1},L_{1})\otimes\mathrm{KhR}_{N}(B_{2},L_{2})$	$\displaystyle\cong\mathrm{KhR}_{N}(\phi(L_{1}))\otimes\mathrm{KhR}_{N}(\phi(L_{2}))$
		$\displaystyle\to\mathrm{KhR}_{N}(\phi(L_{1}\sqcup L_{2}))$
		$\displaystyle\cong\mathrm{KhR}_{N}(B_{1}\#_{\partial}B_{2},L_{1}\sqcup L_{2})$

where the map in the second line comes from the Künneth theorem (this is guaranteed to be an isomorphism when working with field coefficients). The compatibility on the level of morphisms is verified similarly. ∎