ScalingStacks

	$\displaystyle\mathrm{Vert}_{n-1}$	$\displaystyle\Longrightarrow$	$\displaystyle\cap_{i,n}\cup_{i,n-1},$
	$\displaystyle\cup_{i,n-1}\cap_{i,n}$	$\displaystyle\Longrightarrow$	$\displaystyle\mathrm{Vert}_{n},$
	$\displaystyle\mathrm{Vert}_{n}$	$\displaystyle\Longrightarrow$	$\displaystyle\cup_{i,n-1}\cap_{i,n},$
	$\displaystyle\cap_{i,n}\cup_{i,n-1}$	$\displaystyle\Longrightarrow$	$\displaystyle\mathrm{Vert}_{n-1}$

induce grading-preserving bimodule homomorphisms

	$\displaystyle H^{n-1}\longrightarrow{\mathcal{F}}(\cap_{i,n})\otimes_{H^{n}}{\mathcal{F}}(\cup_{i,n-1})\{1\},$		(1)
	$\displaystyle{\mathcal{F}}(\cup_{i,n-1})\otimes_{H^{n-1}}{\mathcal{F}}(\cap_{i,n})\{1\}\longrightarrow H^{n},$		(2)
	$\displaystyle H^{n}\longrightarrow{\mathcal{F}}(\cup_{i,n-1})\otimes_{H^{n-1}}{\mathcal{F}}(\cap_{i,n})\{-1\},$		(3)
	$\displaystyle{\mathcal{F}}(\cap_{i,n})\otimes_{H^{n}}{\mathcal{F}}(\cup_{i,n-1})\{-1\}\longrightarrow H^{n-1},$		(4)

(we used that ${\mathcal{F}}(\mathrm{Vert}_{n})\cong H^{n},{\mathcal{F}}(\cup_{i,n-1}\cap_{i,n})\cong{\mathcal{F}}(\cup_{i,n-1})\otimes_{H^{n-1}}{\mathcal{F}}(\cap_{i,n}),$ etc.) Isotopies between compositions of these cobordisms translate into relations between homomorphisms. These relations imply that the functors of tensoring with ${\mathcal{F}}(\cap_{i,n})$ and ${\mathcal{F}}(\cup_{i,n-1})$ are biadjoint, up to grading shifts. Precisely, let $F_{\cup}$ be the functor of tensoring with ${\mathcal{F}}(\cup_{i,n-1})$ and $F_{\cap}$ the functor of tensoring with ${\mathcal{F}}(\cap_{i,n})$ (viewed as functors between categories of $H^{n}$ and $H^{n-1}$-modules).