ScalingStacks

Proof. The first claim follows from the fact that $i^{*}M\simeq 0$ if and only if for all $F_{R}$-modules $N$, $\mathrm{Map}_{F^{\infty}_{R}}(i_{!}N,M)\simeq 0$. In turn, this holds if and only if for any map of $F^{\infty}_{R}$-modules $Q\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}P$ with cofiber of the form $i_{!}N$,

$$\mathrm{Map}_{F^{\infty}_{R}}(P,M)\simeq\mathrm{Map}_{F^{\infty}_{R}}(Q,M).$$

The second claim follows from the fact that, if the cofiber of $f$ lies in the essential image of $i_{!}$, then for any local object $L$, $\mathrm{Map}_{F^{\infty}_{R}}(M^{\prime},L)\simeq\mathrm{Map}_{F^{\infty}_{R}}(M,L)$. ∎