ScalingStacks

Proof. The spectral Yoneda embedding

$$\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})\simeq\mathrm{N}(\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf})$$

is adjoint to a simplicial functor

$$\mathfrak{C}[\mathcal{C}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf})$$

which evidently factors through the full simplicial subcategory $\Omega^{\infty}\Upsilon(\mathcal{C})$ spanned by the stably representable functors. The map $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))$ is the adjoint of the resulting map $\mathfrak{C}[\mathcal{C}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}\Upsilon(\mathcal{C})$.

To see that this map is an equivalence, we observe first that it is essentially surjective: indeed, a stably representable cofibrant and fibrant functor $X\colon\mathfrak{C}[\mathcal{C}]^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ is necessarily of the form $X\simeq\mathrm{Map}(-,A)$ for some spectrum object $A=\{a_{i}\}$ of $\mathcal{C}\simeq\mathrm{N}(\mathfrak{C}[\mathcal{C}]^{\mathrm{fib}})$. Since $\mathcal{C}$ is stable, $a_{i}\simeq\Sigma^{i}a$ for some object $a$ of $\mathcal{C}$, so $X$ is in the image of $\mathcal{C}$ (which sends $a$ to the presheaf represented by $\Sigma^{\infty}a$). This map is also fully faithful, because if $a$ and $b$ are any pair of objects of $\mathcal{C}$, then

$$\map(\Sigma^{\infty}b,\Sigma^{\infty}a)\simeq\map(b,\Omega^{\infty}\Sigma^{\infty}a)\simeq\map(b,a)$$

since $a\simeq\Omega^{\infty}\Sigma^{\infty}a$. ∎