ScalingStacks

Lemma 8.1.7. If $\xi^{-1}(M)$ has no maximum, then the construction above gives an isomorphism

$$E_{\xi}\wedge_{{\mathcal{S}}^{\bullet}_{M}(Z)}M_{f}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}M_{f}\wedge_{{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})}E_{f\circ\xi},$$

and $f$ provides a morphism of bimodule $2$-representations $L^{\bullet}_{f\circ\xi}\to L^{\bullet}_{\xi}$.