ScalingStacks

Remark 4.3.3. Assume $E_{1}$ admits a left adjoint $F_{1}$. The data of the map $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ corresponds by adjunction to the data of a map

$$\lambda:F_{1}E_{2}\xrightarrow{\bullet\eta_{1}}F_{1}E_{2}E_{1}F_{1}\xrightarrow{F_{1}\sigma F_{1}}F_{1}E_{1}E_{2}F_{1}\xrightarrow{\varepsilon_{1}\bullet}E_{2}F_{1}.$$

The commutativity of the diagrams (4.3.1) is equivalent to the commutativity of the diagrams (4.2.1). Assume the diagrams commute. We obtain a lax bi-$2$-representation $(E_{i,j})$ on ${\mathcal{W}}$ (cf §4.2.1).

Let $(m,\varsigma)\in\Delta_{E}{\mathcal{W}}$. We have an adjunction isomorphism

$$\phi:\operatorname{Hom}\nolimits(E_{2}(m),E_{1}(m))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits(F_{1}E_{2}(m),m).$$

Let $\pi=\phi^{-1}(\varsigma)\in Z\operatorname{Hom}\nolimits(E_{2}(m),E_{1}(m))$. The object $(m,\pi)$ is in $\Delta_{\sigma}{\mathcal{W}}$ and $(m,\varsigma)\mapsto(m,\pi)$ defines a fully faithful functor of differential categories $\Delta_{E}{\mathcal{W}}\to\Delta_{\sigma}{\mathcal{W}}$.

Assume now $\lambda$ is invertible. The canonical map $f_{i}:(E_{0,1}E_{1,0})^{i}\to E_{i,i}$ is invertible. Let $\varsigma_{i}=b_{i}\circ f_{i}^{-1}$. Consider $r\in\{1,\ldots,i-1\}$. We have

	$\displaystyle\varsigma_{i}\circ(T_{r}\otimes 1)$	$\displaystyle=b_{r-1}\circ(E_{01}E_{10})^{r-1}(\varsigma_{2}\circ(T_{1}\otimes 1)\circ f_{2})\circ(E_{01}E_{10})^{r+1}b_{i-r-1}\circ f_{i}^{-1}$
		$\displaystyle=b_{r-1}\circ(E_{01}E_{10})^{r-1}(\varsigma_{2}\circ(1\otimes T_{1})\circ f_{2})\circ(E_{01}E_{10})^{r+1}b_{i-r-1}\circ f_{i}^{-1}$
		$\displaystyle=\varsigma_{i}\circ(1\otimes T_{r})$

As a consequence, the functor above is an isomorphism of differential categories $\Delta_{E}{\mathcal{W}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\sigma}{\mathcal{W}}$.