ScalingStacks

Proof. Certainly essential surjectivity is determined on the level of the homotopy category, so it suffices to show that, for all pairs of objects $a$ and $b$ of ${\mathcal{A}}$, $\pi_{n}\mathrm{Map}(a,b)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{n}\mathrm{Map}(fa,fb)$ for all integers $n$ whenever this is the case for $n=0$. Since ${\mathcal{A}}$ and ${\mathcal{B}}$ are stable,

$$\pi_{0}\mathrm{Map}(\Sigma^{n}a,b)\cong\pi_{n}\mathrm{Map}(a,b)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{n}\mathrm{Map}(fa,fb)\cong\pi_{0}\mathrm{Map}(\Sigma^{n}fa,fb),$$

so this is immediate. ∎