ScalingStacks

Proof. First note that $M\eta_{{\mathcal{A}}}\colon M{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M^{2}{\mathcal{A}}$ sends $x\colon\widehat{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ to the functor $\widehat{\eta_{{\mathcal{A}}}}_{!}x\colon\widehat{M{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ induced by homotopy left Kan extension along $\widehat{\eta_{{\mathcal{A}}}}:\widehat{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{M{\mathcal{A}}}$. If $x=\mathrm{Map}_{A}(-,a)$ is represented by the object $a$ of ${\mathcal{A}}$, then the universal properties of representable functors and homotopy left Kan extensions force an equivalence $\widehat{\eta_{{\mathcal{A}}}}_{!}x\cong\mathrm{Map}(-,\eta_{{\mathcal{A}}}(a))$, so that $\widehat{\eta_{{\mathcal{A}}}}_{!}x$ is represented by $\eta_{{\mathcal{A}}}(a)$. It follows that the restrictions of $\eta_{M{{\mathcal{A}}}}$ and $M\eta_{{\mathcal{A}}}$ to ${\mathcal{A}}$ are equivalent. ∎