ScalingStacks

Proof. We start with the equivalence $\left(1\right)\iff\left(2\right)$. The first part follows from the fact that $\Phi$ is a monomorphism and the second part follows from the fact that $\Phi$ is a homotopy equivalence of Kan complexes. In the equivalence $\left(3\right)\iff\left(4\right)$, the first part follows from the fact that $\Sigma A\to J\left(A\right)$ is an epimorphism and the second part can be seen as follows: the maps $\overline{f},\overline{g}\colon J\left(B\right)\to\mathcal{C}$ are homotopic rel $J\left(A\right)$ if and only if they are equivalent as elements of the $\infty$-category that is the fiber over $\overline{f}|_{J\left(A\right)}=\overline{g}|_{J\left(A\right)}$ (which is also a homotopy fiber) of the categorical fibration $\mathcal{C}^{J\left(B\right)}\to\mathcal{C}^{J\left(A\right)}$. Since we have functorial categorical equivalences $\Sigma\left(A\right)\overset{\sim}{\longrightarrow}J\left(A\right)$ and $\Sigma\left(B\right)\overset{\sim}{\longrightarrow}J\left(B\right)$, this is the same as showing that the corresponding maps $\overline{F},\overline{G}\colon\Sigma\left(B\right)\to\mathcal{C}$ are equivalent in the fiber of $\mathcal{C}^{\Sigma\left(B\right)}\to\mathcal{C}^{\Sigma\left(A\right)}$ (which is also the homotopy fiber). This in turn is the same as having $\overline{F},\overline{G}$ homotopic rel. $\Sigma A$. It is left to show the equivalence $\left(2\right)\iff\left(4\right)$. The first part is clear. The second part amounts to showing the equivalence of two extension problems. If $F|_{A}=G|_{A}$, we get a map $F\cup_{A}G$ from $B\cup_{A}B\simeq B\rtimes_{A}\partial\Delta^{1}$ to $\hom_{\mathcal{C}}^{M}\left(X,Y\right)$ and $F$ and $G$ are homotopic rel. $A$ if and only if $F\cup_{A}G$ extends to the relative cylinder $B\rtimes_{A}\Delta^{1}$. In terms of maps to $\mathcal{C}$, this is equivalent to the extension problem

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On the other hand, from $\overline{F}|_{\Sigma A}=\overline{G}|_{\Sigma A}$ we get a map $\overline{F}\cup_{\Sigma A}\overline{G}$ from $\Sigma B\rtimes_{\Sigma A}\partial\Delta^{1}$ to $\mathcal{C}$ and $\overline{F}$ and $\overline{G}$ are homotopic rel. $\Sigma A$ if and only if it extends to the relative cylinder $\Sigma B\rtimes_{\Sigma A}\Delta^{1}$. By 2.16 for $D=\Delta^{1},\partial\Delta^{1}$, the two extension problems are isomorphic. ∎