ScalingStacks

Lemma 8.1.8. If $\xi^{-1}(M)$ has no maximum, then the construction above gives an isomorphism

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

and $f^{\#}$ provides a morphism of bimodule $2$-representations $L_{\xi}\to L_{f\circ\xi}$.